A vector is something that has both a magnitude and a direction. Something which only has a magnitude is a scalar. In these notes a vector is represented by a bold letter, such as A or B. In the textbook, or when written by hand, a vector is shown with an arrow on top. Writing A or B (not bold, without an arrow) means just the magnitude of the vector.

What are some examples of vectors?

What are some examples of scalars?

• Examples of vectors: velocity, acceleration, force
• Examples of scalars: mass, distance, temperature

In a picture, a vector is shown as an arrow pointing in the direction of a vector. The length of the arrow is proportional to the magnitude of the vector. Note that multiplying the vector by a negative number reverses the direction of the vector.

A unit vector is:

• a vector with a length of one unit
• very convenient for specifying directions

A unit vector is written as the vector symbol with a ^ on top, like this: . This is spoken as "r-hat".

Three very special unit vectors are , , .

is a unit vector in the x direction.

is a unit vector in the y direction.

is a unit vector in the z direction.

Splitting a vector into components is done using the geometry of a right-angled triangle. The x-component of a vector tells us how much of that vector is in the x-direction. The y-component tells us how much of the vector is in the y-direction.

Take these positive directions: +x = right and +y = up. A vector v points down and to the right. Make v the hypotenuse of a right-angled triangle, with the other two sides parallel to the coordinate axes.

• Use geometry to find the magnitude of each component
• Use the diagram to tell you the sign

cos(θ)

=

vx v

, so

vx = v cos(θ)

Including the positive direction from the diagram, v_x = +v cos(θ)

sin(θ)

=

vy v

, so

vy = v sin(θ)

Including the negative direction from the diagram, v_y = -v sin(θ)

If v = 3.5 m/s and θ = 25 degrees, then:

v_x = 3.5 cos(25) = 3.2 m/s

v_y = -3.5 sin(25) = -1.5 m/s

The entire vector v can then be written in unit vector notation:

v = 3.2 -1.5 m/s

Remember that the component on the side of the triangle adjacent to the angle goes with the cosine; the component opposite the angle goes with the sine.