(a) vector A = 3i + 4j and vector B = 12i - 5j where i and j are unit vectors along X

Question

(a) \(\vec A = \ 3\hat{i} + 4\hat{j}\) and \(\vec B = 12\hat{i} - 5\hat{j}\) where \(\hat{i}\) and \(\hat{j}\) are unit vectors along X and Y axes respectively. Find:

(i) magnitude of \(\vec A\) and (ii) value of \(\vec A.\vec B\).

(b) If \(\vec A = 2\hat{i} + 3\hat{j} - 5\hat{k}\), then find the magnitude of it.

Answer 1

\((a) \ \vec A = 3\hat{i} + 4\hat{j}\ and\ \vec B = 12\hat{i} - 5\hat{j}\) 

\(\text{(i)} \ |\vec A| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16 } = \sqrt{25} = 5 \)

∴ A = 5 unit

\(\vec A.\vec B = (3\hat{i} + 4\hat{j}) .(12\hat{i} - 5\hat{j})\)

\( = 36\hat{i}.\hat{i} - 15\hat{i}\ \hat{j} + 48\hat{j}.\hat{i} - 20\hat{j}.\hat{j}\)

= 36 - 0 + 0 - 20

\(\text{(b)}\ \vec A = 2\hat{i} + 3\hat{j} - 5\hat{k}\) 

\(\therefore \ |\vec A| = \sqrt{(2)^2 + (3)^2 + (-5)^2} = \sqrt{4+ 9 +25 } = \sqrt{38}\) 

or A = \(\sqrt{38}\) unit