What will be the value of α if vectors vector A = (2i + 3j + 8k) and vector B = -4i - 4j + ak

Question

What will be the value of α if vectors\(\vec A = (2\hat{i} + 3\hat{j} + 8\hat{k})\ and\ \vec B = -4\hat{i} - 4\hat{j} + a\hat{k}\) be perpendicular?

(a) -1

(b) \(\frac{1}{2}\)

(c) \(-\frac{1}{2}\)

(d) 1

Answer 1

\(\vec A = 2\hat{i} + 3\hat{j} + 8\hat{k}\ and\ \vec B = -4\hat{i} - 4\hat{j} + \alpha\hat{k}\) 

\(\therefore\ \vec A.\vec B = (2\hat{i} + 3\hat{j} + 8\hat{k}).(-4\hat{i} - 4\hat{j} + \alpha\hat{k})\)

= -8 + 12 + 8α

= 4 + 8α

If \(\vec A\ \bot\vec B\ \)then \(\vec A.\vec B = 0\)

\(\therefore\ 4 + 8\alpha = 0\Rightarrow\alpha = -\frac{1}{2}\)

Thus, option (c) is correct.