Given the vectors A = 2 i + 3 j + 4 k and B = 4 i + 3 j + 2 k , determine the cross product A × B .

Correct answer: C. -6 i + 12 j - 6 k The cross product of two vectors A and B is given by the determinant of their components: A × B = vmatrix i and j and k 2 and 3 and 4 4 and 3 and 2 vmatrix Expanding the determinant along the first row: A × B = i (3 × 2 - 4 × 3) - j (2 × 2 - 4 × 4) + k (2 × 3 - 3 × 4) A × B = i (6 - 12) - j (4 - 16) + k (6 - 12) A × B = -6 i + 12 j - 6 k

Concepts Of Physics MCQ Edition [Volume 1]

Exam: Concepts Of Physics MCQ Edition [Volume 1]
Subject: Physics
Chapter: Physics and Mathematics

Concepts Of Physics MCQ Edition [Volume 1] · Physics · Physics and Mathematics

Given the vectors A = 2 i + 3 j + 4 k and B = 4 i + 3 j + 2 k ,...

Given the vectors

\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}

and

\vec{B} = 4\hat{i} + 3\hat{j} + 2\hat{k}

, determine the cross product

\vec{A} \times \vec{B}

Options

A $6\hat{i} + 12\hat{j} + 6\hat{k}$
B $6\hat{i} - 12\hat{j} + 6\hat{k}$
C $-6\hat{i} + 12\hat{j} - 6\hat{k}$
D $-6\hat{i} - 12\hat{j} - 6\hat{k}$

Correct answer

C. $-6\hat{i} + 12\hat{j} - 6\hat{k}$

Step-by-step solution

The cross product of two vectors

\vec{A}

and

\vec{B}

is given by the determinant of their components:

\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} \text{ and } \hat{j} \text{ and } \hat{k} \\ 2 \text{ and } 3 \text{ and } 4 \\ 4 \text{ and } 3 \text{ and } 2 \end{vmatrix}

Expanding the determinant along the first row:

\vec{A} \times \vec{B} = \hat{i}(3 \times 2 - 4 \times 3) - \hat{j}(2 \times 2 - 4 \times 4) + \hat{k}(2 \times 3 - 3 \times 4)

\vec{A} \times \vec{B} = \hat{i}(6 - 12) - \hat{j}(4 - 16) + \hat{k}(6 - 12)

\vec{A} \times \vec{B} = -6\hat{i} + 12\hat{j} - 6\hat{k}

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