Vectors

A vector is a quantity and direction of avariable, such as; displacement, velocity,

acceleration and force.

A vector is represented graphically asan arrow. The length represents the

magnitude (quantity) of the vector, the angleshows the direction.

Vectors can be multiplied or divided by a scalar.

If A = 5 m

A scalar is a value with no direction

then 2A is:

(10 m)

Vectors can be added to find the “net” resultant

A

B

Connect vectors “tip to tail”

A + B

To find the magnitude of the resultant vector, use the Pythagorean Theorem a2 + b2 = c2

To find the direction of the vector, use trig.

functions

tan = opp/adj sin = opp/hyp

cos = adj/hyp

6 m/s2

1.5 m/s2

magnitude: c2 = a2 + b2 = (6 m/s2)2 + (1.5 m/s2)2

c = 6.18 m/s2

direction: tan = opp/adj = 1.5 m/s2 / 6 m/s2 = .25

= tan-1 (.25) = 14°

Measure all directions from the positive x-axis

If no logical “x-axis” exists, explain yourcoordinate system. (i.e.: 20° ahead of downthe field)

A vector can be broken down into its x and ycomponents

38°

x comp

y comp5 m

cos 38° = x comp / 5 m x = (5 m) cos38° = 3.94 m

y = (5 m) sin38° = 3.08 m

You can add vectors that are not perpendicular by adding their x and y components to make a right triangle,

and find the resultant

5 m

75°

25°

x = 5 m(cos 75°) + 4 m(cos 25°)

4 m

= 4.92 m

4.92 m

y = 5 m(sin75°) + 4 m(sin25°)= 6.52 m

6.52 m

magnitude2 = (4.92 m)2 + (6.52 m)2 mag = 8.17 m

direction: tan = 6.52/4.92 = 53°

Find the resultant vector’s magnitude and direction

29°

5 N

19°

7.5 N

3 N