Grade 12 Math: Projections Worksheet

Projections in Geometry

Read each question carefully and answer to the best of your ability. Show all your work for full credit.

1. Define what a scalar projection is and explain its geometric interpretation.

2. Differentiate between a scalar projection and a vector projection. Provide an example where each would be used.

3. The projection of vector \(\vec{a}\) onto vector \(\vec{b}\) is given by the formula: proj\(_\vec{b}\vec{a}\) =  .

4. If two vectors are orthogonal, their scalar projection onto each other is  .

5. Given vectors \(\vec{u} = \langle 3, -2, 5 \rangle\) and \(\vec{v} = \langle 1, 4, -1 \rangle\), calculate the scalar projection of \(\vec{u}\) onto \(\vec{v}\).

6. Using the same vectors from question 5, calculate the vector projection of \(\vec{u}\) onto \(\vec{v}\).

7. For two non-zero vectors \(\vec{a}\) and \(\vec{b}\), if proj\(_\vec{b}\vec{a}\) = \(\vec{0}\), what can be concluded about the relationship between \(\vec{a}\) and \(\vec{b}\)?

8. The magnitude of the vector projection of \(\vec{a}\) onto \(\vec{b}\) is always equal to the absolute value of the scalar projection of \(\vec{a}\) onto \(\vec{b}\).

9. A force \(\vec{F} = \langle 4, 6 \rangle\) N is applied to an object, causing a displacement \(\vec{d} = \langle 7, 1 \rangle\) m. Calculate the work done by the force. (Hint: Work done is the scalar projection of the force onto the displacement, multiplied by the magnitude of the displacement.)