Question

If the vectors \( \vec{a} = 2\vec{i} + 3\vec{j} - \vec{k} \), \( \vec{b} = 4\vec{i} - \vec{j} + 3\vec{k} \), and \( \vec{c} = p\vec{i} + \vec{j} - \vec{k} \) are coplanar, then: \[ |\vec{a} \times \vec{c}| = ? \]

• A. \( \sqrt{14} \)
• B. \( \frac{3\sqrt{10}}{2} \)
• C. \( \sqrt{26} \)
• D. \( \frac{\sqrt{90}}{4} \)

Hint:

For coplanar vectors, always check the scalar triple product. For cross product calculations, use the determinant form.

Answer 1

The Correct Option is B

Step 1: Use the condition of coplanarity.
We are given that \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are coplanar. The condition for coplanarity is that the scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \). We need to compute \( |\vec{a} \times \vec{c}| \), so we first compute the cross product \( \vec{a} \times \vec{c} \).
Step 2: Calculate the cross product \( \vec{a} \times \vec{c} \).
The cross product is computed as: \[ \vec{a} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
2 & 3 & -1
p & 1 & -1 \end{vmatrix} \] This simplifies to: \[ \vec{a} \times \vec{c} = (3 - (-1)) \hat{i} - (2 - (-1)) \hat{j} + (2 - 3p) \hat{k} \] \[ = 4\hat{i} - 3\hat{j} + (2 - 3p)\hat{k} \]
Step 3: Find the magnitude of the cross product.
The magnitude is given by: \[ |\vec{a} \times \vec{c}| = \sqrt{(4)^2 + (-3)^2 + (2 - 3p)^2} \] After simplification, we get: \[ |\vec{a} \times \vec{c}| = \frac{3\sqrt{10}}{2} \]