Geometric Vectors

A Vector is denoted by a symbol . This denotes vector v.

Every vector has a magnitude or length. This magnitude is always positive. We can measure the magnitude with a ruler, or given the coordinates of the vector we can use the distance formula to calculate the magnitude.

The amplitude is the directed angle between the x – axis and the vector.

amplitude

The sum of two or more vectors is called the resultant.

Find

Two vectors are opposites if they have the same magnitude and opposite directions.

Parallel vectors have the same or opposite direction.

Vectors have a horizontal and vertical component.

Where y is the vertical component and x is the

horizontal component.

What we create with the horizontal and vertical components is a right triangle. We can use our knowledge of right triangle trigonometry to figure out the sides and the amplitude.

Example:

Given a vector with a magnitude of 4.3cm and an amplitude of 51°, find the vertical and horizontal components.

4.3cm 4.3

51° 51°

To find the vertical component we have:

To find the horizontal component:

Example:

Two forces are acting on an object. One is 7N (Newtons) east and the other is 24N south. What is the magnitude of the resulting force upon the object? What is the direction of the resulting force on the object?

7N

q

24N

To find the magnitude of the resulting force, we can use the Pythagorean Theorem. Since what we have are two sides of a right triangle. The magnitude of the third side or is then:

49 + 756 = c

625 = c

25N = c

So our resulting magnitude is 25N.

To find the amplitude then we can use the tangent function. We have:

or q = 73.74°