3.4 Unit Vectors and Vector Algebra

The vectors $(1, + x$ -direction) and $(1, + y$ -direction), shown in FIGURE 3.16, have some interesting and useful properties. Each has a magnitude of 1, has no units, and is parallel to a coordinate axis. A vector with these properties is called a unit vector. These unit vectors have the special symbols

\begin{array}{l} î \equiv (1, positive x - direction) \\ \hat{j} \equiv (1, positive y - direction) \end{array}

The notation $î$ (read “i hat”) and $ĵ$ (read “j hat”) indicates a unit vector with a magnitude of 1. Recall that the symbol $\equiv$ means “is defined as.”

The figure illustrates unit vectors using the x y coordinate plane.

FIGURE 3.16 Full Alternative Text

Unit vectors establish the directions of the positive axes of the coordinate system. Our choice of a coordinate system may be arbitrary, but once we decide to place a coordinate system on a problem we need something to tell us “That direction is the positive x-direction.” This is what the unit vectors do.

The unit vectors provide a useful way to write component vectors. The component vector $\vec{A}_{x}$ is the piece of vector $\vec{A}$ that is parallel to the x-axis. Similarly, $\vec{A}_{y}$ is parallel to the y-axis. Because, by definition, the vector $î$ points along the x-axis and $ĵ$ points along the y-axis, we can write

\begin{array}{l} {\vec{A}}_{x} = A_{x} î \\ {\vec{A}}_{y} = A_{y} \hat{j} \end{array}

(3.6)

Equations 3.6 separate each component vector into a length and a direction. The full decomposition of vector $\vec{A}$ can then be written

\vec{A} = \vec{A}_{x} + \vec{A}_{y} = A_{x} î + A_{y} ĵ

(3.7)

FIGURE 3.17 shows how the unit vectors and the components fit together to form vector $\vec{A}$ .

The figure illustrates the components of a displacement vector labeled vector “A” using unit vectors on the x y coordinate plane.

FIGURE 3.17 Full Alternative Text

NOTE In three dimensions, the unit vector along the $+ z - direction$ is called $\hat{k}$ , and to describe vector $\vec{A}$ we would include an additional component vector $\vec{A}_{z} = A_{z} \hat{k}$ .

EXAMPLE 3.5 Run rabbit run!

A rabbit, escaping a fox, runs $40.0 °$ north of west at $10.0 m / s$ . A coordinate system is established with the positive x-axis to the east and the positive y-axis to the north. Write the rabbit’s velocity in terms of components and unit vectors.

VISUALIZE FIGURE 3.18 shows the rabbit’s velocity vector and the coordinate axes. We’re showing a velocity vector, so the axes are labeled $v_{x}$ and $v_{y}$ rather than x and y.

The figure illustrates the components of a velocity vector labeled vector v on the x y coordinate plane.

FIGURE 3.18 Full Alternative Text

SOLVE $10.0 m / s$ is the rabbit’s speed, not its velocity. The velocity, which includes directional information, is

\vec{v} = (10.0 m / s, 40.0 ° n o r t h o f w e s t)

Vector $\vec{v}$ points to the left and up, so the components $v_{x}$ and $v_{y}$ are negative and positive, respectively. The components are

\begin{array}{l} v_{x} = - (10.0 m/s) cos 40.0 ° = - 7.66 m / s \\ v_{y} = + (10.0 m/s) sin 40.0 ° = 6.43 m / s \end{array}

With $v_{x}$ and $v_{y}$ now known, the rabbit’s velocity vector is

\vec{v} = v_{x} î + v_{y} ĵ = (- 7.66 î + 6.43 ĵ) m / s

Notice that we’ve pulled the units to the end, rather than writing them with each component.

REVIEW Notice that the minus sign for $v_{x}$ was inserted manually. Signs don’t occur automatically; you have to set them after checking the vector’s direction.

3.4 Unit Vectors and Vector Algebra

FIGURE 3.16 The unit vectors $î$ and $ĵ$ .

FIGURE 3.17 The decomposition of vector $\vec{A}$ is $A_{x} î + A_{y} ĵ$ .

EXAMPLE 3.5 Run rabbit run!

FIGURE 3.18 The velocity vector $\vec{v}$ is decomposed into components $v_{x}$ and $v_{y}$ .