Skip to content
Skip to main content

About this free course

Download this course

Share this free course

Introducing vectors for engineering applications
Introducing vectors for engineering applications

Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.

2.1 Horizontal and vertical components

Described image
Figure 13 A vector and its components

Vectors are intuitively described using magnitude and direction, but they are more usefully described according to horizontal and vertical components, as illustrated in Figure 13.

A vector that can be described by a magnitude and a direction can also be described by component vectors with:

  • the horizontal component pointing in the direction of the positive x-axis
  • the vertical component pointing in the direction of the positive y-axis.

Together, a vector and its component vectors form a right-angled triangle, and the magnitudes of the three vectors define the lengths of the edges of the triangle. So, referring to Figure 13, we can use the sine and cosine functions to determine the magnitudes of the component vectors. The vector bold v has magnitude absolute value of v and direction theta, and from these its components are calculated as follows.

The sine function is defined as

sine equals opp divided by hyp comma

and by recognising that the vertical component is opposite the angle theta, we get

sine of theta equals vertical divided by absolute value of v or vertical equals absolute value of v times sine of theta full stop

Similarly, the cosine function is defined as

cosine equals adj divided by hyp comma

and by recognising that the horizontal component is adjacent to the angle theta, we get

cosine of theta equals horizontal divided by absolute value of v or horizontal equals absolute value of v times cosine of theta full stop

Activity 7

The vector bold v has a magnitude of 4 makes an angle of 30° with the positive x-axis, as shown in the following diagram. Identify the magnitudes of the horizontal and vertical components of bold v.