Vector Components | Calculus III (2024)

Working with vectors in a plane is easier when we are working in a coordinate system. When the initial points and terminal points of vectors are given in Cartesian coordinates, computations become straightforward.

We have seen how to plot a vector when we are given an initial point and a terminal point. However, because a vector can be placed anywhere in a plane, it may be easier to perform calculations with a vector when its initial point coincides with the origin. We call a vector with its initial point at the origin a standard-position vector. Because the initial point of any vector in standard position is known to be [latex](0,0)[/latex], we can describe the vector by looking at the coordinates of its terminal point. Thus, if vector [latex]\bf{v}[/latex] has its initial point at the origin and its terminal point at [latex](x,y)[/latex], we write the vector in component form as

When a vector is written in component form like this, the scalars [latex]x[/latex] and [latex]y[/latex] are called the components of [latex]\bf{v}[/latex].

Recall that vectors are named with lowercase letters in bold type or by drawing an arrow over their name. We have also learned that we can name a vector by its component form, with the coordinates of its terminal point in angle brackets. However, when writing the component form of a vector, it is important to distinguish between [latex]\langle x,y \rangle[/latex] and [latex](x,y)[/latex]. The first ordered pair uses angle brackets to describe a vector, whereas the second uses parentheses to describe a point in a plane. The initial point of [latex]\langle x,y \rangle[/latex] is [latex](0,0)[/latex]; the terminal point of [latex]\langle x,y \rangle[/latex] is [latex](x,y)[/latex].

When we have a vector not already in standard position, we can determine its component form in one of two ways. We can use a geometric approach, in which we sketch the vector in the coordinate plane, and then sketch an equivalent standard-position vector. Alternatively, we can find it algebraically, using the coordinates of the initial point and the terminal point. To find it algebraically, we subtract the [latex]x[/latex]-coordinate of the initial point from the [latex]x[/latex]-coordinate of the terminal point to get the [latex]x[/latex] component, and we subtract the [latex]y[/latex]-coordinate of the initial point from the [latex]y[/latex]-coordinate of the terminal point to get the [latex]y[/latex] component.

To find the magnitude of a vector, we calculate the distance between its initial point and its terminal point. The magnitude of vector
[latex]\bf{v}[/latex][latex]=\langle x,y\rangle[/latex] is denoted [latex]\bf{||v||}[/latex], or [latex]\bf{|v|}[/latex], and can be computed using the formula

.

Note that because this vector is written in component form, it is equivalent to a vector in standard position, with its initial point at the origin and terminal point [latex](x,y)[/latex]. Thus, it suffices to calculate the magnitude of the vector in standard position. Using the distance formula to calculate the distance between initial point [latex](0,0)[/latex] and terminal point [latex](x,y)[/latex], we have

Based on this formula, it is clear that for any vector [latex]\bf{v}[/latex], [latex]\bf{||v||}[/latex][latex] \geq 0[/latex], and [latex]\bf{||v||}[/latex][latex] = 0[/latex] if and only if [latex]\bf{v}[/latex][latex] = 0[/latex].

The magnitude of a vector can also be derived using the Pythagorean theorem, as in the following figure.

We have defined scalar multiplication and vector addition geometrically. Expressing vectors in component form allows us to perform these same operations algebraically.

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

Now that we have established the basic rules of vector arithmetic, we can state the properties of vector operations. We will prove two of these properties. The others can be proved in a similar manner.

Proof of Cummutative Property

Let [latex]{\bf{u}}=\langle x_1, y_1\rangle[/latex] and [latex]{\bf{v}}=\langle x_2, y_2\rangle[/latex]. Apply the commutative property for real numbers:

[latex]_\blacksquare[/latex]

Proof of Distrubutive Property

Apply the distributive property for real numbers:

[latex]\begin{array}{ccc} \hfill r({\bf{u}} + {\bf{v}}) & =\hfill & r \cdot \langle x_1 + x_2,y_1 + y_2\rangle \hfill \\ \hfill & =\hfill & \langle r(x_1 + x_2), r(y_1 + y_2)\rangle \hfill \\ \hfill & =\hfill & \langle rx_1 + rx_2, ry_1 + ry_2\rangle \hfill \\ \hfill & =\hfill & \langle rx_1,ry_1\rangle \langle rx_2, ry_2\rangle \hfill \\ \hfill & =\hfill & r{\bf{u}} + r{\bf{v}}\end{array}[/latex]

[latex]_\blacksquare[/latex]

We have found the components of a vector given its initial and terminal points. In some cases, we may only have the magnitude and direction of a vector, not the points. For these vectors, we can identify the horizontal and vertical components using trigonometry (Figure 16).

Consider the angle [latex]\theta[/latex] formed by the vector [latex]{\bf{v}}[/latex] and the positive [latex]x[/latex]-axis. We can see from the triangle that the components of vector [latex]{\bf{v}}[/latex] are [latex]\langle ||{\bf{v}}||\cos{\theta},||{\bf{v}}||\sin{\theta} \rangle[/latex]. Therefore, given an angle and the magnitude of a vector, we can use the cosine and sine of the angle to find the components of the vector.