Question

ii. Find the values of \( c \) which satisfy \( |c\vec{u}| = 3 \) where \( \vec{u} = \hat{i} + 2\hat{j} + 3\hat{k} \).

Hint:

To find the magnitude of a vector, use \( |\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \) where \( v_1, v_2, v_3 \) are the components of the vector.

Answer 1

The magnitude of the vector \( \vec{u} = \hat{i} + 2\hat{j} + 3\hat{k} \) is given by: \[ |\vec{u}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} \] Now, we are given that \( |c \vec{u}| = 3 \). So, \[ |c \vec{u}| = |c| \cdot |\vec{u}| = 3 \] \[ |c| \cdot \sqrt{14} = 3 \] \[ |c| = \frac{3}{\sqrt{14}} \] Thus, the values of \( c \) are: \[ c = \pm \frac{3}{\sqrt{14}} \]