7.5. Vector Projection http://www.ck12.org
Example C
Find the vector projection of vectorv=< 3 , 4 >onto vectoru=< 5 ,− 12 >
Solution: Since the scalar projection has already been found in Example B, you should multiply the scalar by the
“onto” unit vector.
−^3313 < 135 ,−^1213 >=<−^165169 ,^396169 >
Concept Problem Revisited
Vector projection is useful in physics applications involving force and work.
When the box is pulled by vectorvsome of the force is wasted pulling up against gravity. In real life this may be
useful because of friction, but for now this energy is inefficiently wasted in the horizontal movement of the box.
Vocabulary
Thevector projectionis the vector produced when one vector is resolved into two component vectors, one that is
parallel to the second vector and one that is perpendicular to the second vector. The parallel vector is the vector
projection.
Thescalar projectionis the length of the vector projection. When the scalar projection is negative it means that the
two vectors are heading in opposite directions and the angle between the vectors is greater than 90◦.
Guided Practice
- Sketch vectors< 1 , 3 >and< 5 , 2 >. What is the vector projection of< 1 , 3 >onto< 5 , 2 >? Sketch the
projection. - Sketch the vector<− 2 ,− 2 >and< 4 ,− 2 >. Explain using a sketch why a negative scalar projection of
<− 2 ,− 2 >onto< 4 ,− 2 >makes sense. - A father is pulling his daughter up a hill. The hill has a 20◦incline. The daughter is on a sled that sits on the
ground and has a rope that the father pulls with a force of 100 lb as he walks. The rope makes a 39◦angle with the
slope. What is the effective force that the father exerts moving his daughter and the sled up the hill?
Answers: