Vector Basics: Learn It 4

Unit Vectors

A unit vector is a vector with magnitude [latex]1[/latex]. Unit vectors are useful because they indicate direction without being tied to a specific magnitude.

For any nonzero vector [latex]\mathbf{v}[/latex], you can find a unit vector [latex]\mathbf{u}[/latex] that points in the same direction by multiplying the vector by the reciprocal of its magnitude:

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

This process is called normalization.

unit vector

A unit vector has magnitude [latex]1[/latex]. To find the unit vector in the direction of [latex]\mathbf{v}[/latex], use:

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

Let’s verify this works:

[latex]||\mathbf{u}|| = \left|\left|\frac{1}{||\mathbf{v}||}\mathbf{v}\right|\right| = \frac{1}{||\mathbf{v}||} \cdot ||\mathbf{v}|| = 1 ✓[/latex]

 

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

  1. Find a unit vector with the same direction as [latex]{\bf{v}}[/latex].
  2. Find a vector [latex]{\bf{w}}[/latex] with the same direction as [latex]{\bf{v}}[/latex] such that [latex]||{\bf{w}}|| = 7[/latex].

Standard Unit Vectors

Sometimes it’s convenient to express vectors in terms of horizontal and vertical components using standard unit vectors:

  • [latex]\mathbf{i} = \langle 1,0 \rangle[/latex] (points along positive [latex]x[/latex]-axis)
  • [latex]\mathbf{j} = \langle 0,1 \rangle[/latex] (points along positive [latex]y[/latex]-axis)
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]i[/latex] and [latex]j[/latex].

Any vector [latex]\mathbf{v} = \langle x,y \rangle[/latex] can be written as a linear combination of [latex]\mathbf{i}[/latex] and [latex]\mathbf{j}[/latex]:

[latex]\mathbf{v} = \langle x,y \rangle = x\langle 1,0 \rangle + y\langle 0,1 \rangle = x\mathbf{i} + y\mathbf{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].
  1. Express the vector [latex]{\bf{w}} = \langle 3,-4 \rangle[/latex] in terms of standard unit vectors.
  2. Vector [latex]{\bf{u}}[/latex] is a unit vector that forms an angle of [latex]60°[/latex] with the positive [latex]x[/latex]-axis. Use standard unit vectors to describe [latex]{\bf{u}}[/latex].