Unit Vectors

Learning Outcomes

  • Express a vector in terms of unit vectors.
  • Give two examples of vector quantities.

Unit Vectors

A unit vector is a vector with magnitude [latex]1[/latex]. For any nonzero vector [latex]{\bf{v}}[/latex], we can use scalar multiplication to find a unit vector [latex]{\bf{u}}[/latex] that has the same direction as [latex]{\bf{v}}[/latex]. To do this, we multiply the vector by the reciprocal of its magnitude:

[latex]{\bf{u}} = \frac{1}{||{\bf{v}}||}{\bf{v}}[/latex].

Recall that when we defined scalar multiplication, we noted that [latex]||k{\bf{v}}|| = |k| \cdot ||{\bf{v}}||[/latex]. For [latex]{\bf{u}} = \frac{1}{||{\bf{v}}||}{\bf{v}}[/latex], it follows that [latex]||{\bf{u}}|| = \frac{1}{||{\bf{v}}||}(||{\bf{v}}||) = 1[/latex]. We say that [latex]{\bf{u}}[/latex] is the unit vector in the direction of [latex]{\bf{v}}[/latex] (Figure 18). The process of using scalar multiplication to find a unit vector with a given direction is called normalization.

This image has two figures. The first is a vector labeled “v.” The second figure is a vector in the same direction labeled “u.” This vector has a length of 1 unit.

Figure 1. The vector [latex]{\bf{v}}[/latex] and associated unit vector [latex]{\bf{u}}=\frac{1}{||{\bf{v}}||}{\bf{v}}[/latex]. In this case, [latex]||{\bf{v}}||>1[/latex].

Example: Finding a Unit Vector

Let [latex]{\bf{v}} = \langle 1,2 \rangle[/latex].

Try It

Let [latex]{\bf{v}} = \langle 9,2 \rangle[/latex]. Find a vector with magnitude [latex]5[/latex] in the opposite direction as [latex]{\bf{v}}[/latex].

We have seen how convenient it can be to write a vector in component form. Sometimes, though, it is more convenient to write a vector as a sum of a horizontal vector and a vertical vector. To make this easier, let’s look at standard unit vectors. The standard unit vectors are the vectors [latex]{\bf{i}} = \langle 1,0 \rangle[/latex] and [latex]{\bf{j}} = \langle 0,1 \rangle[/latex] (Figure 2.18).

This figure has the x and y axes of a coordinate system in the first quadrant. On the x-axis there is a vector labeled “i,” which equals <1,0>. The second vector is on the y-axis and is labeled “j” which equals <0,1>.

Figure 2. The standard unit vectors [latex]{\bf{i}}[/latex] and [latex]{\bf{j}}[/latex].

By applying the properties of vectors, it is possible to express any vector in terms of [latex]{\bf{i}}[/latex] and [latex]{\bf{j}}[/latex] in what we call a linear combination:

[latex]||{\bf{v}}|| = \langle x,y \rangle = \langle x,0 \rangle + \langle 0,y \rangle = x\langle 1,0 \rangle + y\langle 0,1 \rangle = x{\bf{i}} + y{\bf{j}}[/latex].

Thus, [latex]{\bf{v}}[/latex] is the sum of a horizontal vector with magnitude [latex]x[/latex], and a vertical vector with magnitude [latex]y[/latex], as in the following figure.

This figure is a right triangle. The horizontal side is labeled “xi.” The vertical side is labeled “yj.” The hypotenuse is a vector labeled “v.”

Figure 3. The vector [latex]{\bf{v}}[/latex] is the sum of [latex]x{\bf{i}}[/latex] and [latex]y{\bf{j}}[/latex].

Example: Using Standard Unit Vectors

Try It

Let [latex]{\bf{a}} = \langle 16,-11 \rangle[/latex] and let [latex]{\bf{b}}[/latex] be a unit vector that forms an angle of [latex]225°[/latex] with the positive [latex]x[/latex]-axis. Express [latex]{\bf{a}}[/latex] and [latex]{\bf{b}}[/latex] in terms of the standard unit vectors.

Watch the following video to see the worked solution to the above Try IT.

Try It

Applications of Vectors

Because vectors have both direction and magnitude, they are valuable tools for solving problems involving such applications as motion and force. Recall the boat example and the quarterback example we described earlier. Here we look at two other examples in detail.

Example: Finding Resultant Force

Jane’s car is stuck in the mud. Lisa and Jed come along in a truck to help pull her out. They attach one end of a tow strap to the front of the car and the other end to the truck’s trailer hitch, and the truck starts to pull. Meanwhile, Jane and Jed get behind the car and push. The truck generates a horizontal force of [latex]300[/latex] lb on the car. Jane and Jed are pushing at a slight upward angle and generate a force of [latex]150[/latex] lb on the car. These forces can be represented by vectors, as shown in Figure 22. The angle between these vectors is [latex]15°[/latex]. Find the resultant force (the vector sum) and give its magnitude to the nearest tenth of a pound and its direction angle from the positive [latex]x[/latex]-axis.

This image is the side view of an automobile. From the front of the automobile there is a horizontal vector labeled “300 pounds.” Also, from the front of the automobile there is another vector labeled “150 pounds.” The angle between the two vectors is 15 degrees.

Figure 5. Two forces acting on a car in different directions.

Example: Finding Resultant Velocity

An airplane flies due west at an airspeed of [latex]425[/latex] mph. The wind is blowing from the northeast at [latex]40[/latex] mph. What is the ground speed of the airplane? What is the bearing of the airplane?

Try It

An airplane flies due north at an airspeed of [latex]550[/latex] mph. The wind is blowing from the northwest at [latex]50[/latex] mph. What is the ground speed of the airplane?

Watch the following video to see the worked solution to the above Try IT.