Let a = i + 4 j + 2 k, b = 3 i - 2 j + 7 k, c = 2 i - j + 4 k. If a vector d satisfies d b = c b and d a = 24, then | d|^2 is equal to:

Given: a = i + 4 j + 2 k, b = 3 i - 2 j + 7 k, c = 2 i - j + 4 k, with d b = c b and d a = 24. Find: | d|^2. From d b = c b, we get ( d - c) b = 0 So, d - c is parallel to b, hence d = c + b Using d a = 24, ( c + b) a = 24 Now, a c = (1)(2) + (4)(-1) + (2)(4) = 2 - 4 + 8 = 6 and a b = (1)(3) + (4)(-2) + (2)(7) = 3 - 8 + 14 = 9 Therefore, 6 + 9 = 24 9 = 18 = 2 So, d = c + 2 b d = (2 i - j + 4 k) + 2(3 i - 2 j + 7 k) = 8 i - 5 j + 18 k Finally, | d|^2 = 8^2 + (-5)^2 + 18^2 = 64 + 25 + 324 = 413 Therefore, the correct value is 413. The solution concludes this value, but the listed page label says Option D, whereas 413 matches option C. Hence the correct option is C.

Let a = i + 4 j + 2 k, b = 3 i - 2 j + 7 k, c = 2 i - j + 4 k. If… — Solution | JEE Mathematics

MCQMediumJEE 2023Cross Product

Let $\mathbf{a} = \hat{i} + 4\hat{j} + 2\hat{k}, \mathbf{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}, \mathbf{c} = 2\hat{i} - \hat{j} + 4\hat{k}$ . If a vector $\mathbf{d}$ satisfies $\mathbf{d} \times \mathbf{b} = \mathbf{c} \times \mathbf{b}$ and $\mathbf{d} \cdot \mathbf{a} = 24$ , then $|\mathbf{d}|^2$ is equal to:

A
$323$
B
$423$
C
$413$
D
$313$

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: $\mathbf{a} = \hat{i} + 4\hat{j} + 2\hat{k}$ , $\mathbf{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ , $\mathbf{c} = 2\hat{i} - \hat{j} + 4\hat{k}$ , with $\mathbf{d} \times \mathbf{b} = \mathbf{c} \times \mathbf{b}$ and $\mathbf{d} \cdot \mathbf{a} = 24$ .

Find: $|\mathbf{d}|^2$ .

From $\mathbf{d} \times \mathbf{b} = \mathbf{c} \times \mathbf{b}$ , we get

(\mathbf{d} - \mathbf{c}) \times \mathbf{b} = \mathbf{0}

So, $\mathbf{d} - \mathbf{c}$ is parallel to $\mathbf{b}$ , hence

\mathbf{d} = \mathbf{c} + \lambda \mathbf{b}

Using $\mathbf{d} \cdot \mathbf{a} = 24$ ,

(\mathbf{c} + \lambda \mathbf{b}) \cdot \mathbf{a} = 24

Now,

\mathbf{a} \cdot \mathbf{c} = (1)(2) + (4)(-1) + (2)(4) = 2 - 4 + 8 = 6

and

\mathbf{a} \cdot \mathbf{b} = (1)(3) + (4)(-2) + (2)(7) = 3 - 8 + 14 = 9

Therefore,

6 + 9\lambda = 24

9\lambda = 18

\lambda = 2

So,

\mathbf{d} = \mathbf{c} + 2\mathbf{b}

\mathbf{d} = (2\hat{i} - \hat{j} + 4\hat{k}) + 2(3\hat{i} - 2\hat{j} + 7\hat{k}) = 8\hat{i} - 5\hat{j} + 18\hat{k}

Finally,

|\mathbf{d}|^2 = 8^2 + (-5)^2 + 18^2 = 64 + 25 + 324 = 413

Therefore, the correct value is $413$ . The solution concludes this value, but the listed page label says Option D, whereas $413$ matches option C. Hence the correct option is C.

Common mistakes

Assuming directly that $\mathbf{d} = \mathbf{c}$ from $\mathbf{d} \times \mathbf{b} = \mathbf{c} \times \mathbf{b}$ is incorrect. Equality of cross products only implies $\mathbf{d} - \mathbf{c}$ is parallel to $\mathbf{b}$ . Use $\mathbf{d} = \mathbf{c} + \lambda \mathbf{b}$ instead.
Missing the sign in the dot product while computing $\mathbf{a} \cdot \mathbf{b}$ or $\mathbf{a} \cdot \mathbf{c}$ leads to a wrong value of $\lambda$ . Carefully include the negative component $-2$ in $\mathbf{b}$ and $-1$ in $\mathbf{c}$ .
Calculating $|\mathbf{d}|^2$ as $|\mathbf{d}|$ is a common error. The question asks for the square of the magnitude, so add the squares of the components without taking the square root.

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