When two vectors are parallel, one of the vectors can be expressed in terms of the other using scalar multiplication.

Question

Given that \( \lvert \vec{v} \rvert = 5 \), find the magnitude of each of the following vectors:

a. \(2\vec{v}\)

b. \(-4\vec{v}\)

c. \(\dfrac{1}{5}\vec{v}\)

Solution

a. \(\lvert 2\vec{v} \rvert\) \(= \lvert 2 \rvert \lvert \vec{v} \rvert\) \( = (2)(5) = 10\)

b. \( \lvert -4\vec{v} \rvert = \lvert -4\rvert \lvert \vec{v} \rvert = (4)(5) = 20 \)

c. \( \left \lvert \dfrac{1}{5}\vec{v} \right \rvert = \left \lvert \dfrac{1}{5} \right \rvert \lvert \vec{v} \rvert = \dfrac{1}{5} (5) = 1 \)

The vector \( \frac{1}{5}\vec{v} \) is a special vector called a unit vector because it has magnitude \(1\).

To create a unit vector in the direction of a non-zero \( \vec{u} \), multiply \( \vec{u} \) by the scalar equal to the reciprocal of the magnitude of \( \vec{u} \).

That is, \( \vec{v} = \dfrac{1}{\lvert \vec{u}\rvert} \vec{u} \) is a unit vector in the direction of \( \vec{u} \).

Notationally, this is written \( \hat{u} = \dfrac{1}{\lvert \vec{u} \rvert} \vec{u} \).

Question

If \( \lvert \vec{a} \rvert = 12 \), state a unit vector in the direction of \( \vec{a} \).

Solution

A unit vector in the direction of \( \vec{a} \) would be the vector \( \vec{a} \) multiplied by a scalar equal to the reciprocal of \( \lvert \vec{a} \rvert \).

That is,

In this module, we have

• distinguished the difference between a vector and a scalar,
• defined opposite vectors, equal vectors, and the magnitude of a vector,
• looked at how to determine the angle between two vectors,
• talked about scalar multiplication, and
• talked about a unit vector.