Question

Let $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - \hat{k}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b} \times \vec{c} = \vec{b} \times \vec{a}$ and $\vec{c} \cdot \vec{a} = 0$, then $\vec{c} \cdot \vec{b}$ is

• A. $\frac{1}{2}$
• B. $\frac{3}{2}$
• C. $\frac{3}{2}$
• D.

Hint:

When given vector equations like $\vec{b} \times \vec{c} = \vec{b} \times \vec{a}$, manipulate them to find relationships between the vectors. This often involves rearranging terms or taking dot products. Remember the property that if $\vec{b} \times (\vec{c} - \vec{a}) = \vec{0}$, then $(\vec{c} - \vec{a})$ must be parallel to $\vec{b}$, i.e., $\vec{c} - \vec{a} = k\vec{b}$ for some scalar $k$. Then use the additional condition, like $\vec{c} \cdot \vec{a} = 0$, to solve for $k$ and ultimately find the required dot product.

Answer 1

The Correct Option is D

Solution not provided in the original text.