DOT PRODUCT AND VECTOR PROJECTION

LT 7.5 – I can represent and operate with vectors algebraically.

A third vector operation The Dot Product

The dot product of v = and w = is defined as

Notice that unlike vector addition and scalar multiplication, the dot product yields a scalar

and not a vector.

Orthogonal Vectors

Vectors a & b are orthogonal if and only if a � b = 0.

Example 1: Find the dot product of u and v. Then determine if u and v are orthogonal.

a. b.

PROPERTIES OF THE DOT PRODUCT If u, v, and w are vectors and k is a scalar,

then the following properties hold:

² Commutative Property u�v = v�u

² Distributive Property u�(v+w) = u�v+u�w

² Scalar Multiplication Property ² k(u�v) = ku�v = u�kv

PROPERTIES OF THE DOT PRODUCT If u, v, and w are vectors and k is a scalar,

then the following properties hold:

² Zero Vector Dot Product Property 0�u = 0

²  Dot Product & Vector Magnitude Relationship ²  u�u = |u|2

EXAMPLE 2 Use the dot product to find the magnitude of

ANGLE BETWEEN VECTORS

If θ is the angle between nonzero vectors a and b then:

EXAMPLE 3:

Find the angle θ between vectors u and v to the nearest tenth of a degree.

VECTOR PROJECTIONS

We have resolved vectors into two perpendicular components, sometimes it is useful for one component to be parallel to another vector.

VECTOR PROJECTIONS Projection of u onto v

Let u and v be nonzero vectors, and let w1 and w2 be vector components of u such that w1 is

parallel to v as shown. Then vector w1 is called the vector projection of u onto v, denoted

EXAMPLE 4: Find the projection of

Then write u as the sum of two orthogonal vectors, one of which is the projection of u

onto v.

The projection of u onto v is a vector parallel to v, this vector will not necessarily have the

same direction as v.

Projection with Direction Opposite v

Find the projection of u onto v, when

Then write u as the sum of two orthogonal vectors, one of which is the projection of u onto v.

Step 1: Find the projection vector.

Step 2: Find w2.

Since u = w1 + w2, then w2 = u – w1

If the vector u represents a force, then projvu represents the effect of that force acting in the

direction of v. Example 5: A 3000 pound car sits on a hill inclined at

30°. Ignoring the force of friction, what force is required to keep the car from rolling down the hill?

The weight of the car is the force exerted due to gravity

To find the force, – w1 required to keep the car from rolling down the hill, project F onto a unit vector in the direction of

the side of the hill.

Step 1: Find a unit vector in the direction of the hill.

Step 2: Find w1 = proj v F.

The force required is –w1 = -(-1500v) = 1500 pounds

CALCULATING WORK

In elementary physics, the formula for calculating work is

W = (amount of force in the direction of AB)(distance)

W = F�AB

F = force vector AB = directed distance

Example 6: A person pushes a car with a constant force of 120 newtons at a constant angle of 45°. Find the work done in joules moving the car 10 meters.