💡Alright, so I've got this problem here that I need to solve. It says that for any integers ( a_1, a_2, a_3, ldots, a_{2n} ), the expression [a_1 a_2 ldots a_{2n} left( a_1^2 - a_2^2 + a_3^2 - a_4^2 + ldots + a_{2n-1}^2 - a_{2n}^2 right)]is divisible by 6. I need to show that this is always true, no matter what integers I choose for ( a_1 ) through ( a_{2n} ).Okay, first off, I remember that to show something is divisible by 6, I need to show that it's divisible by both 2 and 3. Since 6 is the product of these two primes, and they're coprime, if the expression is divisible by both, it's divisible by 6. So, I'll tackle this in two parts: divisibility by 2 and divisibility by 3.Starting with divisibility by 2. Let's look at the expression:[a_1 a_2 ldots a_{2n} times text{(something)}]If any of the ( a_i ) is even, then the product ( a_1 a_2 ldots a_{2n} ) is even, and hence the entire expression is even. So, if there's at least one even number among the ( a_i ), we're good for divisibility by 2.But what if all the ( a_i ) are odd? Then, each ( a_i ) is odd, so ( a_i^2 ) is also odd. Now, looking at the expression inside the parentheses:[a_1^2 - a_2^2 + a_3^2 - a_4^2 + ldots + a_{2n-1}^2 - a_{2n}^2]Since each ( a_i^2 ) is odd, subtracting two odds gives an even number. For example, ( a_1^2 - a_2^2 ) is even because odd minus odd is even. Similarly, each pair ( a_{2k-1}^2 - a_{2k}^2 ) is even. So, the entire sum inside the parentheses is a sum of even numbers, which is even. Therefore, even if all ( a_i ) are odd, the expression inside the parentheses is even, making the entire expression ( N ) even. So, regardless of whether the ( a_i ) are even or odd, ( N ) is divisible by 2.Alright, that takes care of divisibility by 2. Now, onto divisibility by 3.Again, looking at the expression:[a_1 a_2 ldots a_{2n} times text{(something)}]If any of the ( a_i ) is divisible by 3, then the product ( a_1 a_2 ldots a_{2n} ) is divisible by 3, and hence ( N ) is divisible by 3. So, if there's at least one multiple of 3 among the ( a_i ), we're good.But what if none of the ( a_i ) is divisible by 3? Well, I remember that any integer is congruent to 0, 1, or 2 modulo 3. Since none of the ( a_i ) is divisible by 3, each ( a_i ) must be congruent to either 1 or 2 modulo 3.Let's consider the expression inside the parentheses:[a_1^2 - a_2^2 + a_3^2 - a_4^2 + ldots + a_{2n-1}^2 - a_{2n}^2]Since each ( a_i ) is either 1 or 2 modulo 3, let's compute ( a_i^2 ) modulo 3:- If ( a_i equiv 1 mod 3 ), then ( a_i^2 equiv 1 mod 3 ).- If ( a_i equiv 2 mod 3 ), then ( a_i^2 equiv 4 equiv 1 mod 3 ).So, regardless of whether ( a_i ) is 1 or 2 modulo 3, ( a_i^2 equiv 1 mod 3 ). Therefore, each term ( a_i^2 ) is congruent to 1 modulo 3.Now, looking at the expression inside the parentheses:[a_1^2 - a_2^2 + a_3^2 - a_4^2 + ldots + a_{2n-1}^2 - a_{2n}^2]Since each ( a_i^2 equiv 1 mod 3 ), each pair ( a_{2k-1}^2 - a_{2k}^2 equiv 1 - 1 equiv 0 mod 3 ). Therefore, each pair cancels out modulo 3, and the entire expression inside the parentheses is congruent to 0 modulo 3. Hence, the expression inside the parentheses is divisible by 3.Therefore, even if none of the ( a_i ) is divisible by 3, the expression inside the parentheses is divisible by 3, making ( N ) divisible by 3.Putting it all together, since ( N ) is divisible by both 2 and 3, it must be divisible by 6. Therefore, the original expression is always divisible by 6, regardless of the integers chosen for ( a_1, a_2, ldots, a_{2n} ).I think that covers both cases for divisibility by 2 and 3. I don't see any gaps in the logic, but let me double-check.For divisibility by 2: If any ( a_i ) is even, done. If all are odd, the expression inside is even, so still divisible by 2.For divisibility by 3: If any ( a_i ) is divisible by 3, done. If none are, then each ( a_i^2 equiv 1 mod 3 ), so the alternating sum becomes a sum of zeros modulo 3, hence divisible by 3.Yes, that seems solid. So, the conclusion is that the expression is divisible by 6.

💡Alright, so I have this problem where I'm given a second-order matrix M, and it satisfies two equations. The first one is that when M multiplies the vector [1, 0], it gives [1, 0]. The second one is that when M multiplies [1, 1], it gives [2, 2]. I need to find the result of M squared multiplied by [1, -1]. Hmm, okay, let's break this down step by step.First, I know that a second-order matrix has four elements, usually represented as:M = [a b] [c d]So, M is a 2x2 matrix with elements a, b, c, d. My goal is to find these elements so that I can construct M, then square it, and finally multiply it by the vector [1, -1].Starting with the first equation: M * [1, 0] = [1, 0]. Let's write this out in terms of matrix multiplication.Multiplying M by [1, 0] would be:[a b] [1] [a*1 + b*0] [a][c d] * [0] = [c*1 + d*0] = [c]And this is supposed to equal [1, 0]. So, that gives us two equations:1) a = 12) c = 0Okay, so from the first equation, I immediately know that a is 1, and c is 0. That simplifies our matrix M to:M = [1 b] [0 d]Now, moving on to the second equation: M * [1, 1] = [2, 2]. Let's perform this multiplication with our current M.Multiplying M by [1, 1]:[1 b] [1] [1*1 + b*1] [1 + b][0 d] * [1] = [0*1 + d*1] = [d]And this should equal [2, 2]. So, we have two more equations:3) 1 + b = 24) d = 2From equation 3, subtracting 1 from both sides gives b = 1. From equation 4, we directly get d = 2.So now, our matrix M is fully determined:M = [1 1] [0 2]Great, so now I have M. The next step is to compute M squared, which is M multiplied by itself. Let's do that.Computing M^2:M^2 = M * M = [1 1] [1 1] [1*1 + 1*0, 1*1 + 1*2] [0 2] * [0 2] = [0*1 + 2*0, 0*1 + 2*2]Calculating each element:First row, first column: 1*1 + 1*0 = 1 + 0 = 1First row, second column: 1*1 + 1*2 = 1 + 2 = 3Second row, first column: 0*1 + 2*0 = 0 + 0 = 0Second row, second column: 0*1 + 2*2 = 0 + 4 = 4So, M squared is:M^2 = [1 3] [0 4]Alright, now I need to compute M^2 multiplied by the vector [1, -1]. Let's set that up.Multiplying M^2 by [1, -1]:[1 3] [1] [1*1 + 3*(-1)] [1 - 3] [-2][0 4] * [-1] = [0*1 + 4*(-1)] = [0 - 4] = [-4]So, the result is [-2, -4].Wait, let me double-check my calculations to make sure I didn't make a mistake.First, M was determined correctly from the given conditions. Then, squaring M, I followed the matrix multiplication rules correctly. Each element was computed step by step, and it seems correct. Then, multiplying M^2 by [1, -1], I did the dot product for each row, which also seems correct.Hmm, everything checks out. So, I think the final result is [-2, -4].**Final Answer**The result of ( M^2 begin{bmatrix} 1 -1 end{bmatrix} ) is boxed{begin{bmatrix} -2 -4 end{bmatrix}}.

💡Okay, so I have this problem with vectors. Let me read it again: Given vectors a = (1, 2) and b = (2, -3). There's another vector c that satisfies two conditions. First, (c + a) is parallel to b, and second, c is perpendicular to (a + b). I need to find vector c.Hmm, okay. Let me break this down. I remember that if two vectors are parallel, one is a scalar multiple of the other. So, if (c + a) is parallel to b, then there must be some scalar k such that c + a = k * b. That makes sense. So, I can write that as an equation.So, c + a = k * b. That means c = k * b - a. Let me write that out. Since a is (1, 2) and b is (2, -3), then c = k*(2, -3) - (1, 2). Let me compute that. Multiplying k into b gives (2k, -3k). Subtracting a from that would be (2k - 1, -3k - 2). So, c is (2k - 1, -3k - 2). Okay, that's the expression for c in terms of k.Now, the second condition is that c is perpendicular to (a + b). I remember that two vectors are perpendicular if their dot product is zero. So, I need to compute the dot product of c and (a + b) and set it equal to zero.First, let me find a + b. Adding vectors a and b component-wise: a is (1, 2) and b is (2, -3). So, a + b = (1 + 2, 2 + (-3)) = (3, -1). Got that.So, now I have c = (2k - 1, -3k - 2) and (a + b) = (3, -1). Their dot product should be zero. Let me compute that.The dot product is (2k - 1)*3 + (-3k - 2)*(-1). Let me calculate each term separately. First term: (2k - 1)*3 = 6k - 3. Second term: (-3k - 2)*(-1) = 3k + 2. So, adding these together: 6k - 3 + 3k + 2.Combine like terms: 6k + 3k is 9k, and -3 + 2 is -1. So, the equation becomes 9k - 1 = 0. Solving for k, I add 1 to both sides: 9k = 1, so k = 1/9.Alright, so k is 1/9. Now, I can substitute this back into the expression for c. Remember, c was (2k - 1, -3k - 2). Let me plug in k = 1/9.First component: 2*(1/9) - 1 = 2/9 - 1. To subtract these, I can write 1 as 9/9, so 2/9 - 9/9 = -7/9.Second component: -3*(1/9) - 2 = -3/9 - 2. Simplify -3/9 to -1/3, so it's -1/3 - 2. Again, write 2 as 6/3, so -1/3 - 6/3 = -7/3. Wait, that doesn't seem right. Let me check my calculation.Wait, no, hold on. The second component is -3k - 2. So, plugging in k = 1/9: -3*(1/9) is -1/3, and then minus 2 is -1/3 - 2. To combine these, 2 is 6/3, so -1/3 - 6/3 is -7/3. Hmm, but in the previous step, I thought it was -20/9. Did I make a mistake?Wait, let me go back. Maybe I miscalculated. Wait, no, in the initial substitution, I had c = (2k - 1, -3k - 2). So, plugging k = 1/9:First component: 2*(1/9) = 2/9, minus 1 is 2/9 - 9/9 = -7/9. That's correct.Second component: -3*(1/9) = -1/3, minus 2 is -1/3 - 2. To combine these, 2 is 6/3, so -1/3 - 6/3 = -7/3. Wait, that's different from what I thought earlier. So, is it -7/3 or -20/9?Wait, maybe I made a mistake in the initial calculation. Let me double-check the dot product step.So, c is (2k - 1, -3k - 2). (a + b) is (3, -1). Their dot product is (2k - 1)*3 + (-3k - 2)*(-1). Let's compute that again.First term: (2k - 1)*3 = 6k - 3.Second term: (-3k - 2)*(-1) = 3k + 2.Adding them together: 6k - 3 + 3k + 2 = 9k -1. So, 9k -1 = 0, so k = 1/9. That seems correct.So, substituting back into c: (2*(1/9) -1, -3*(1/9) -2) = (2/9 - 9/9, -1/3 - 6/3) = (-7/9, -7/3). Wait, so that would be (-7/9, -7/3). But in my initial thought process, I thought it was (-7/9, -20/9). Hmm, that's inconsistent.Wait, maybe I made a mistake in the second component. Let me compute it again.Second component: -3k - 2. k = 1/9. So, -3*(1/9) = -1/3. Then, -1/3 - 2. To combine these, convert 2 to thirds: 2 = 6/3. So, -1/3 - 6/3 = (-1 -6)/3 = -7/3. So, that's correct.But in the initial problem, the user wrote the final answer as (-7/9, -20/9). So, that's conflicting with my current calculation. Did I make a mistake?Wait, let me check the expression for c again. c = k*b - a. So, k*b is (2k, -3k). Then, subtracting a, which is (1, 2), gives (2k -1, -3k -2). That seems correct.Wait, but when I compute the second component, I get -3k -2. So, with k = 1/9, that's -1/3 -2. Which is -7/3. But in the initial problem, the answer was (-7/9, -20/9). So, perhaps I made a mistake in the dot product.Wait, let me recompute the dot product step.c is (2k -1, -3k -2). (a + b) is (3, -1). So, the dot product is (2k -1)*3 + (-3k -2)*(-1).Compute each term:(2k -1)*3 = 6k -3.(-3k -2)*(-1) = 3k + 2.Adding them together: 6k -3 + 3k +2 = 9k -1.Set equal to zero: 9k -1 =0 => k=1/9.That seems correct.So, then c = (2*(1/9)-1, -3*(1/9)-2) = (2/9 -9/9, -1/3 -6/3) = (-7/9, -7/3).Wait, so why does the initial problem say (-7/9, -20/9)? Maybe I made a mistake in the initial problem's answer.Wait, let me check the initial problem again. The user wrote:"Given vectors a = (1, 2), b = (2, -3), if vector c satisfies (c + a) parallel to b and c perpendicular to (a + b), then c = (quad)."Then, the user's initial thought process led to c = (-7/9, -20/9). But according to my calculation, it's (-7/9, -7/3). Which is (-7/9, -21/9). Hmm, that's different.Wait, so perhaps I made a mistake in the expression for c. Let me double-check.c = k*b - a.b is (2, -3). So, k*b is (2k, -3k). Then, subtracting a, which is (1, 2), gives (2k -1, -3k -2). That seems correct.Wait, but when I compute the second component: -3k -2. If k=1/9, then -3*(1/9) is -1/3, and -1/3 -2 is -7/3, which is -21/9. So, that would be (-7/9, -21/9). But the initial answer was (-7/9, -20/9). So, there's a discrepancy.Wait, perhaps I made a mistake in the dot product calculation. Let me check that again.Dot product: (2k -1, -3k -2) · (3, -1) = (2k -1)*3 + (-3k -2)*(-1).Compute each term:(2k -1)*3 = 6k -3.(-3k -2)*(-1) = 3k + 2.Adding them: 6k -3 + 3k +2 = 9k -1.Set to zero: 9k -1 =0 => k=1/9.That seems correct.So, substituting k=1/9 into c:First component: 2*(1/9) -1 = 2/9 -9/9 = -7/9.Second component: -3*(1/9) -2 = -1/3 -2 = -7/3 = -21/9.So, c is (-7/9, -21/9). But the initial answer was (-7/9, -20/9). Hmm, so there must be a mistake somewhere.Wait, maybe I made a mistake in the expression for c. Let me check.c = k*b - a.b is (2, -3). So, k*b is (2k, -3k). Then, subtracting a, which is (1, 2), gives (2k -1, -3k -2). That seems correct.Wait, but perhaps I made a mistake in the dot product. Let me check the components again.c is (2k -1, -3k -2). (a + b) is (3, -1). So, the dot product is (2k -1)*3 + (-3k -2)*(-1).Compute each term:(2k -1)*3 = 6k -3.(-3k -2)*(-1) = 3k + 2.Adding them: 6k -3 + 3k +2 = 9k -1.Set to zero: 9k -1 =0 => k=1/9.That seems correct.So, substituting k=1/9 into c:First component: 2*(1/9) -1 = 2/9 -9/9 = -7/9.Second component: -3*(1/9) -2 = -1/3 -2 = -7/3 = -21/9.So, c is (-7/9, -21/9). But the initial answer was (-7/9, -20/9). So, perhaps the initial answer was incorrect.Alternatively, maybe I made a mistake in the expression for c. Let me check again.Wait, perhaps I made a mistake in the expression for c. Let me think.Given that (c + a) is parallel to b, so c + a = k*b.Therefore, c = k*b - a.Yes, that's correct.So, c = k*(2, -3) - (1, 2) = (2k -1, -3k -2). That seems correct.Wait, but when I plug in k=1/9, I get (-7/9, -21/9). So, perhaps the initial answer was wrong.Alternatively, maybe I made a mistake in the dot product.Wait, let me compute the dot product again.c = (2k -1, -3k -2).(a + b) = (3, -1).Dot product: (2k -1)*3 + (-3k -2)*(-1).Compute:(2k -1)*3 = 6k -3.(-3k -2)*(-1) = 3k + 2.Total: 6k -3 + 3k +2 = 9k -1.Set to zero: 9k -1 =0 => k=1/9.That seems correct.So, substituting k=1/9, c = (2*(1/9) -1, -3*(1/9) -2) = (2/9 -9/9, -1/3 -6/3) = (-7/9, -7/3).Which is (-7/9, -21/9). So, the initial answer of (-7/9, -20/9) must be incorrect.Alternatively, perhaps I made a mistake in the initial problem's setup.Wait, let me check the problem again."Given vectors a = (1, 2), b = (2, -3), if vector c satisfies (c + a) parallel to b and c perpendicular to (a + b), then c = (quad)."So, the problem is correct. So, perhaps the initial answer was wrong.Alternatively, maybe I made a mistake in the calculation.Wait, let me compute c again.c = k*b - a.k=1/9.So, k*b = (2*(1/9), -3*(1/9)) = (2/9, -1/3).Subtracting a: (2/9 -1, -1/3 -2) = (-7/9, -7/3).Yes, that's correct.So, the correct answer should be (-7/9, -7/3), which is (-7/9, -21/9). So, perhaps the initial answer was a typo or mistake.Alternatively, maybe I made a mistake in the dot product.Wait, let me compute the dot product with c = (-7/9, -21/9) and (a + b) = (3, -1).Dot product: (-7/9)*3 + (-21/9)*(-1) = (-21/9) + (21/9) = 0. Okay, that's correct.So, the dot product is zero, which satisfies the perpendicular condition.Also, c + a = (-7/9 +1, -21/9 +2) = (2/9, -3/9) = (2/9, -1/3). Which is (2/9, -1/3). Is this parallel to b?b is (2, -3). So, is (2/9, -1/3) a scalar multiple of (2, -3)?Let's see: 2/9 divided by 2 is 1/9. -1/3 divided by -3 is 1/9. So, yes, (2/9, -1/3) = (1/9)*(2, -3). So, that's correct.So, everything checks out. Therefore, the correct answer is (-7/9, -7/3), which is (-7/9, -21/9). So, the initial answer of (-7/9, -20/9) must be incorrect.Wait, but in the initial problem, the user wrote the final answer as (-7/9, -20/9). So, perhaps they made a mistake. Alternatively, maybe I made a mistake in my calculation.Wait, let me check the second component again.c = (2k -1, -3k -2). k=1/9.So, second component: -3*(1/9) -2 = -1/3 -2.Convert 2 to thirds: 2 = 6/3.So, -1/3 -6/3 = -7/3. Which is -21/9.So, that's correct.Wait, maybe the initial problem had a different vector a or b? Let me check.The problem says a = (1, 2), b = (2, -3). So, that's correct.Hmm, I think the initial answer was wrong. So, the correct answer is (-7/9, -21/9), which simplifies to (-7/9, -7/3).But let me write it as (-7/9, -21/9) to have a common denominator.So, in conclusion, the correct vector c is (-7/9, -21/9).

💡First, I need to determine the total number of eggs in the pack. I know there are 3 rotten eggs, and the probability of selecting 2 rotten eggs is given as 0.0047619047619047615.I'll start by setting up the probability equation. The probability of choosing the first rotten egg is 3 divided by the total number of eggs, which I'll denote as ( n ). After selecting one rotten egg, there are now 2 rotten eggs left out of ( n - 1 ) total eggs. So, the probability of choosing a second rotten egg is ( frac{2}{n - 1} ).Multiplying these probabilities together gives the overall probability of selecting 2 rotten eggs:[frac{3}{n} times frac{2}{n - 1} = 0.0047619047619047615]Next, I'll simplify this equation:[frac{6}{n(n - 1)} = 0.0047619047619047615]To solve for ( n ), I'll rearrange the equation:[n(n - 1) = frac{6}{0.0047619047619047615} approx 1260]Now, I need to find two consecutive integers whose product is approximately 1260. Calculating the square root of 1260 gives me an approximate midpoint of 35.5. Testing the integers around this midpoint, I find that:[35 times 36 = 1260]Therefore, the total number of eggs in the pack is 36.

💡First, I need to determine the total number of possible outfits without any restrictions. This is calculated by multiplying the number of shirts, pants, and hats: 7 shirts × 5 pants × 7 hats = 245 total outfits.Next, I need to find out how many of these outfits have the shirt and hat of the same color. Since there are 7 colors available for both shirts and hats, and 5 options for pants, the number of outfits with matching shirt and hat colors is 7 colors × 5 pants = 35 outfits.Finally, to find the number of valid outfits where the shirt and hat are different colors, I subtract the number of outfits with matching colors from the total number of outfits: 245 total outfits - 35 matching color outfits = 210 valid outfits.

💡Alright, so I need to find the equation of a plane that contains a specific point and a given line. The point is (4, 2, -1), and the line is described by the equations:[frac{x - 3}{4} = frac{y + 1}{-1} = frac{z - 2}{2}]Hmm, okay. I remember that to define a plane, I need a point and a normal vector. Since the plane contains the given point, that's one point. But I also need a normal vector. To find the normal vector, I can use the direction vector of the line and another vector that lies on the plane.First, let me write down the parametric equations of the line. The given line has direction ratios 4, -1, 2. So, the parametric equations can be written as:[x = 3 + 4t][y = -1 - t][z = 2 + 2t]where ( t ) is a parameter.Now, since the plane contains this line, any point on the line should satisfy the plane equation. Also, the plane contains the point (4, 2, -1). So, I can use these two points to find another vector that lies on the plane.Let me find two points on the line. Let's choose ( t = 0 ) and ( t = 1 ).When ( t = 0 ):[x = 3 + 4(0) = 3][y = -1 - 0 = -1][z = 2 + 2(0) = 2]So, one point on the line is (3, -1, 2).When ( t = 1 ):[x = 3 + 4(1) = 7][y = -1 - 1 = -2][z = 2 + 2(1) = 4]So, another point on the line is (7, -2, 4).Now, I have three points:1. (4, 2, -1) – the given point on the plane.2. (3, -1, 2) – a point on the line.3. (7, -2, 4) – another point on the line.I can use these three points to find two vectors that lie on the plane. Let's call the vectors **v1** and **v2**.Vector **v1** can be found by subtracting the coordinates of (3, -1, 2) from (4, 2, -1):[v1 = (4 - 3, 2 - (-1), -1 - 2) = (1, 3, -3)]Vector **v2** can be found by subtracting the coordinates of (3, -1, 2) from (7, -2, 4):[v2 = (7 - 3, -2 - (-1), 4 - 2) = (4, -1, 2)]Wait a second, that's interesting. Vector **v2** is exactly the direction vector of the line, which makes sense because the line lies on the plane.Now, to find the normal vector **n** to the plane, I can take the cross product of vectors **v1** and **v2**.Let me compute the cross product:[n = v1 times v2 = begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} 1 & 3 & -3 4 & -1 & 2 end{vmatrix}]Calculating the determinant:[n = mathbf{i}(3 cdot 2 - (-3) cdot (-1)) - mathbf{j}(1 cdot 2 - (-3) cdot 4) + mathbf{k}(1 cdot (-1) - 3 cdot 4)][n = mathbf{i}(6 - 3) - mathbf{j}(2 + 12) + mathbf{k}(-1 - 12)][n = 3mathbf{i} - 14mathbf{j} - 13mathbf{k}]So, the normal vector **n** is (3, -14, -13).Now, the general equation of a plane is:[a(x - x_0) + b(y - y_0) + c(z - z_0) = 0]where (a, b, c) is the normal vector and (x₀, y₀, z₀) is a point on the plane. Let's use the point (4, 2, -1).Plugging in the values:[3(x - 4) - 14(y - 2) - 13(z + 1) = 0]Expanding this:[3x - 12 - 14y + 28 - 13z - 13 = 0]Combine like terms:[3x - 14y - 13z + ( -12 + 28 - 13 ) = 0][3x - 14y - 13z + 3 = 0]Wait, that doesn't seem right. Let me check my calculations again.First, cross product:[n = v1 times v2 = begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} 1 & 3 & -3 4 & -1 & 2 end{vmatrix}]Calculating each component:- i component: (3)(2) - (-3)(-1) = 6 - 3 = 3- j component: -( (1)(2) - (-3)(4) ) = -(2 + 12) = -14- k component: (1)(-1) - (3)(4) = -1 - 12 = -13So, n = (3, -14, -13). That seems correct.Now, plane equation:3(x - 4) -14(y - 2) -13(z + 1) = 0Expanding:3x - 12 -14y +28 -13z -13 = 0Combine constants:-12 +28 = 16; 16 -13 = 3So, 3x -14y -13z +3 = 0Hmm, but let me verify if this plane actually contains the given point and the line.First, check the point (4, 2, -1):3(4) -14(2) -13(-1) +3 = 12 -28 +13 +3 = 0Yes, 12 -28 is -16, -16 +13 is -3, -3 +3 is 0. So, it satisfies the equation.Now, check if the line lies on the plane. Let's take the parametric equations:x = 3 +4t, y = -1 -t, z = 2 +2tPlug into the plane equation:3(3 +4t) -14(-1 -t) -13(2 +2t) +3 = 0Compute each term:3*3 =9, 3*4t=12t-14*(-1)=14, -14*(-t)=14t-13*2=-26, -13*2t=-26tSo, putting it all together:9 +12t +14 +14t -26 -26t +3 = 0Combine like terms:Constants: 9 +14 -26 +3 = (9+14)=23; (23-26)= -3; (-3 +3)=0Variables: 12t +14t -26t = (12+14)=26t; 26t -26t=0So, 0 +0 =0. Perfect, it satisfies the equation.Wait, but in my initial calculation, I got 3x -14y -13z +3 =0, but in the previous assistant's answer, the equation was 4x +3y +4z =42. These are different. Did I make a mistake?Wait, let me double-check the cross product.v1 = (1,3,-3)v2 = (4,-1,2)Cross product:i: 3*2 - (-3)*(-1) =6 -3=3j: -(1*2 - (-3)*4)= -(2 +12)= -14k:1*(-1) -3*4= -1 -12= -13So, n=(3,-14,-13). That seems correct.So, the plane equation is 3x -14y -13z +3=0.But the assistant's answer was 4x +3y +4z=42. Maybe I did something wrong.Wait, perhaps I made a mistake in choosing the vectors. Let me try another approach.Another way to find the plane is to use the point and the line. Since the plane contains the line, the direction vector of the line is (4,-1,2). Also, the plane contains the point (4,2,-1). So, I can find another vector from (4,2,-1) to a point on the line, say (3,-1,2).So, vector from (4,2,-1) to (3,-1,2) is (3-4, -1-2, 2 - (-1))= (-1,-3,3)So, vectors on the plane are direction vector of line (4,-1,2) and vector (-1,-3,3). Then, cross product of these two vectors will give the normal vector.Compute cross product:n = (4,-1,2) × (-1,-3,3)= |i j k| |4 -1 2| |-1 -3 3|= i[(-1)(3) - (2)(-3)] - j[(4)(3) - (2)(-1)] + k[(4)(-3) - (-1)(-1)]= i[-3 +6] - j[12 +2] + k[-12 -1]= i[3] - j[14] + k[-13]So, n=(3,-14,-13). Same as before.So, equation of plane is 3x -14y -13z + D=0. Plug in (4,2,-1):3*4 -14*2 -13*(-1) + D=012 -28 +13 + D=0(12-28)= -16; (-16+13)= -3; -3 + D=0 => D=3So, equation is 3x -14y -13z +3=0.But the assistant's answer was 4x +3y +4z=42. These are different. Maybe the assistant made a mistake.Alternatively, perhaps I need to present it differently. Let me see.Alternatively, maybe the assistant used a different method, like finding two equations from the line and combining them.Wait, looking back at the assistant's thought process, they started by expressing the line in parametric form, then tried to relate the equations, leading to two equations: 4x + y =11 and y +2z=3. Then, they tried to combine these into a plane equation.But when they plugged in the point (4,2,-1), they got 4*4 +2=18 and 2 +2*(-1)=0, which didn't satisfy the equations, so they adjusted them.But actually, the correct approach is to find a plane that contains both the point and the line, which requires using the normal vector method.So, perhaps the assistant made a mistake in their approach.Therefore, my conclusion is that the correct plane equation is 3x -14y -13z +3=0.But let me check if this can be simplified or written differently.Alternatively, maybe the assistant's answer is correct, and I made a mistake.Wait, let me check if the plane 4x +3y +4z=42 contains the point (4,2,-1):4*4 +3*2 +4*(-1)=16 +6 -4=18≠42. So, it doesn't satisfy. Therefore, the assistant's answer is incorrect.Hence, my plane equation 3x -14y -13z +3=0 is correct.But let me see if I can write it in a different form.Alternatively, I can write it as 3x -14y -13z = -3.But usually, we prefer the constant term to be positive, so maybe multiply both sides by -1:-3x +14y +13z =3.But that's still different from the assistant's answer.Alternatively, perhaps I made a mistake in choosing the vectors.Wait, another approach: the plane contains the line, so it must contain all points on the line. Therefore, the plane can be expressed as a linear combination of the line's direction vector and the vector from a point on the line to the given point.But I think my earlier method is correct.Alternatively, maybe the assistant used a different method, like finding two points on the plane and then finding the normal vector.Wait, let me try that.Given the line, I can find two points on the line: when t=0, (3,-1,2); when t=1, (7,-2,4). Also, the given point is (4,2,-1).So, three points: (3,-1,2), (7,-2,4), (4,2,-1).Find vectors from (3,-1,2) to (7,-2,4): (4,-1,2)From (3,-1,2) to (4,2,-1): (1,3,-3)Then, cross product of these vectors is the normal vector.Which is exactly what I did earlier, resulting in (3,-14,-13).So, the plane equation is 3x -14y -13z +3=0.Therefore, I think the assistant's answer is incorrect, and my plane equation is correct.But to be thorough, let me check if the plane 4x +3y +4z=42 contains the line.Take a general point on the line: (3+4t, -1 -t, 2 +2t)Plug into 4x +3y +4z:4*(3+4t) +3*(-1 -t) +4*(2 +2t)= 12 +16t -3 -3t +8 +8t= (12-3+8)+(16t-3t+8t)=17 +21tSet equal to 42:17 +21t=42 =>21t=25 =>t=25/21So, only when t=25/21 does the point lie on the plane. Therefore, the plane does not contain the entire line, only a specific point on the line. Hence, the assistant's answer is incorrect.Therefore, my conclusion is that the correct plane equation is 3x -14y -13z +3=0.