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. A corresponding statement applies to the y-component.

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) = v_x / v, so v_x = v cos(θ)

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

sin(θ) = v_y / v, so v_y = 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.

One way to multiply two vectors is to take the dot product, which results in a scalar. Dot products are important in a case such as work, when the work done by a force in moving an object depends on the component of the force in the direction of the displacement.

W = F · d = F d cos(θ)

Note that the dot product is ...

• zero when the vectors are perpendicular to one another
• maximum when they are parallel
• positive when the angle between the vectors is < 90
• negative when the angle between them is > 90

c = a · b = a_x b_x + a_y b_y + a_z b_z

A second way to multiply vectors is to take the cross product, which results in a vector perpendicular to the two vectors in the cross product. An example is a torque:

τ = r × F

The magnitude of the resultant vector is r F sin(θ).

The direction is given by the right-hand rule. Using your right hand, point your fingers in the direction of the first vector (r), curl them into the direction of the second vector (F), and your thumb, sticking out, will point in the direction of the resultant vector.

Note that the cross product is ...

• zero when the vectors are parallel
• maximum when the vectors are perpendicular to one another

Note also that a × b = - b × a.

c = a × b = (a_y b_z - b_y a _z) + (a_z b_x - b_z a_x) + (a_x b_y - b_x a_y)