Question

suppose you multiply vector u by the negative scalar value -0.5. without performing this scalar multiplication, predict the magnitude and direction of the resulting vector, -0.5u, based on your observations from the earlier example.

Explanation:

Step1: Recall Scalar Multiplication Rules

Scalar multiplication of a vector \(\mathbf{u}\) by a scalar \(k\) (here \(k = -0.5\)) affects the magnitude and direction. The magnitude of \(k\mathbf{u}\) is \(|k| \times \text{magnitude of } \mathbf{u}\). The direction is reversed if \(k\) is negative (since negative scalar flips the direction) and scaled by \(|k|\).

Step2: Analyze Magnitude

For \(k=-0.5\), \(|k| = 0.5\). So the magnitude of \(-0.5\mathbf{u}\) is \(0.5\) times the magnitude of \(\mathbf{u}\) (it is halved, as \(0.5=\frac{1}{2}\)).

Step3: Analyze Direction

Since the scalar \(k = -0.5\) is negative, the direction of \(-0.5\mathbf{u}\) is opposite to the direction of \(\mathbf{u}\).

Answer:

The magnitude of \(-0.5\mathbf{u}\) is half (or \(0.5\) times) the magnitude of \(\mathbf{u}\), and the direction of \(-0.5\mathbf{u}\) is opposite to the direction of \(\mathbf{u}\).