Question

If the vectors $2\hat{i} + \hat{j} - a\hat{k}$ and $\hat{i} + 4\hat{j} + \hat{k}$ are perpendicular, find the value of $a$.

Hint:

For perpendicular vectors → always use $\vec{u}\cdot \vec{v} = 0$.

Answer 1

Step 1: Recall condition for perpendicular vectors. Two vectors $\vec{u}$ and $\vec{v}$ are perpendicular if: \[ \vec{u}\cdot \vec{v} = 0 \]

Step 2: Write the given vectors. \[ \vec{u} = 2\hat{i} + \hat{j} - a\hat{k}, \vec{v} = \hat{i} + 4\hat{j} + \hat{k} \]

Step 3: Take the dot product. \[ \vec{u}\cdot \vec{v} = (2)(1) + (1)(4) + (-a)(1) \] \[ = 2 + 4 - a = 6 - a \]

Step 4: Apply perpendicular condition. \[ 6 - a = 0 \implies a = 6 \]

Final Answer: \[ \boxed{a = 6} \]