Fall 2014
Math 151
Week-in-Review 2
1
1. Find the dot product for the following vectors. Are the vectors parallel, perpendicular or
neither? If neither, find the angle between the two vectors.
a. 6, 0 and 5, 3
b. 2, −1 and −4, 2
c. 2, 6 and 1, −3
2. A wagon is pulled a distance of 100 meters along a horizontal path by a constant force of
50 N. If the handle of the wagon is at an angle of 30 0 above the horizontal, how much
work is done?
3. Find the unit vector(s) orthogonal to the vector 3𝑖 + 𝑗
a. What are the unit vector(s) that are parallel to above vector?
4. What value(s) of x will make the following vectors orthogonal?
a. 𝑥𝑖 − 2𝑗 and 𝑥𝑖 + 8𝑗
b. 𝑥, 𝑥 and 1, 𝑥
5. Given the following vectors 𝑎 = 3, −1 and 𝑏 = 2, 3 , find
a. The scalar and vector projection of 𝑎 onto 𝑏
b. The scalar and vector projection of 𝑏 onto 𝑎
6. Under what circumstances would compab = compba for 2 non zero vectors a and b?
7. Given three points of a triangle 𝑃 0, 1 , 𝑄 2, 1 and 𝑅(5, 4), find the angles of the three
vertices of the triangle to the nearest degree.
8. Find the distance of the point (2, 3) from the line 3𝑥 − 4𝑦 + 5 = 0.
9. For what value of c will the angle between the vectors 1, 1 and 1, 𝑐 equal 600 ?
10. Find the vector equation of the line 2𝑦 + 3𝑥 = 5
11. Find the vector equation, Cartesian equation and parametric equations of a line that
passes through the points (1, 3) and is parallel to the vector 1, −2
12. Eliminate the parameter to find the Cartesian equation. Sketch the curve.
a. 𝑥 = sin 𝜃 + tan 𝜃 , 𝑦 = cos 𝜃
b. 𝑥 = 2𝑡 − 1, 𝑦 = 𝑡 2 − 1
c. 𝑥 = 𝑡, 𝑦 = 1 − 𝑡
Fall 2014
Math 151
Week-in-Review 2
2
13. The position of an object moving in the xyplane, after t seconds is given by
𝑟 𝑡 = (𝑡 + 4)𝑖 + (𝑡 2 + 2)𝑗.
a. Find the position of the object after time 𝑡 = 2
b. At what time will the object reach a point
i. (7, 11)?
ii. (9, 12)?
c. Find the Cartesian equation.
14. Are the following pairs of lines parallel, perpendicular or neither? If they are not parallel,
find the point of intersection between them
a. 𝑟 𝑡 = 2 + 3𝑡 𝑖 + 4𝑡 − 1 𝑗 and 𝑠 𝑡 = 1 + 4𝑡 𝑖 + (2 − 2𝑡)𝑗
b. 𝑟 𝑡 = −4 + 2𝑡 𝑖 + 5 + 𝑡 𝑗 and 𝑠 𝑡 = 2 + 3𝑡 𝑖 + (4 − 6𝑡)𝑗
15.
16. Given that: