Question

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

Hint:

For vector-scalar equations, check if magnitude is implied; compute vector magnitude using \( \sqrt{x^2 + y^2 + z^2} \).

Answer 1

The vector \( \mathbf{u} = (1, 2, 3) \). The equation \( c \mathbf{u} = 3 \) is ambiguous, as a scalar times a vector cannot equal a scalar. Assume it means the magnitude of the scaled vector equals 3: \[ |c \mathbf{u}| = 3. \] Compute the magnitude of \( \mathbf{u} \): \[ |\mathbf{u}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}. \] Then: \[ |c| \cdot |\mathbf{u}| = 3 \quad \Rightarrow \quad |c| \cdot \sqrt{14} = 3 \quad \Rightarrow \quad |c| = \frac{3}{\sqrt{14}}. \] Thus: \[ c = \pm \frac{3}{\sqrt{14}}. \]