Vector Projections

Tutorial on Vector Projections

Definition

Vector projection is the process of projecting one vector onto another. The projection of vector a onto vector b is given by the formula:

proj_b a = (a · b) / (b · b) * b

Steps to Calculate Vector Projection

1. Calculate the Dot Product: Find a · b.

2. Calculate the Magnitude Squared: Find b · b.

3. Apply the Formula: Substitute the values into the projection formula.

Sample Problems

Problem 1: Project vector a = (3, 4) onto vector b = (1, 2).

1. Calculate a · b = 31 + 42 = 11.

2. Calculate b · b = 11 + 22 = 5.

3. Apply the formula:

   proj_b a = (11/5) * b = (11/5, 22/5)

Problem 2: Project vector a = (2, 3, 4) onto vector b = (1, 0, 0).

1. Calculate a · b = 21 + 30 + 4*0 = 2.

2. Calculate b · b = 11 + 00 + 0*0 = 1.

3. Apply the formula:

   proj_b a = 2 * b = (2, 0, 0)

Here are a few word problems that require vector projections to solve:

1. Force on an Object: A force of 10 N is applied at an angle of 30° to the horizontal. Find the horizontal component of the force using vector projection.

2. Boat Crossing a River: A boat is heading directly across a river with a velocity of 5 m/s. The current flows downstream at 3 m/s. Determine the projection of the boat's velocity vector onto the direction of the riverbank.

3. Shadow Length: A light source is positioned above a hill. If the hill is represented by a vector and the light source by another, find the projection of the light vector onto the hill vector to determine the length of the shadow cast.

4. Work Done: A worker pushes a crate with a force vector of 15 N at an angle of 45° to the direction of motion. Calculate the work done by finding the projection of the force vector onto the direction of motion.