Test 3 Review: Vectors - Annapolis High School



Assessment 3 Review

Unit 2: Vectors

IB HL2

Answer the following:

1. Let u = 2i – 7j + 4k and v = 3i + 3j – 2k. Find the angle between u and v.

2. Find the unit vector in the direction of -5i + 12j.

3. Find the angle between the lines [pic]and [pic].

4. A vector equation of a line is[pic]. Find the equation of this line in the form ax + by = c, where a, b, and c are integers.

5. Find the point of intersection for the following lines: [pic] and [pic].

6. Find a vector perpendicular to the two vectors:

[pic] = [pic] – 3[pic] + 2[pic]

[pic] = –2[pic] + [pic] – [pic]

If [pic] and [pic] are position vectors for the points P and Q, use your answer to part (a), or otherwise, to find the area of the triangle OPQ.

7. Consider the points A(l, 2, 1), B(0, –1, 2), C(1, 0, 2), and D(2, –1, –6).

a. Calculate [pic] × [pic].

b. Find a set of parametric equations for the line through the point D and

perpendicular to the plane P.

c. Find a unit vector which is perpendicular to the plane P.

8. The coordinates of the points P, Q, R and S are (4,1,–1), (3,3,5), (1,0, 2c), and (1,1,2), respectively.

a. Find the value of c so that the vectors [pic] and [pic] are orthogonal.

For the remainder of the question, use the value of c found in part (a) for the coordinate of the point R.

b. Evaluate [pic] × [pic].

c. Find an equation of the line l which passes through the point Q and is parallel to the vector PR.

9. Find the equation of the plane through (1, 2, 3) parallel to 3x + 4y – 5z = 0.

10. Find the equation of the plane through the three points (1, 1, 0), (1, 2, 1) and (-2, 2, -1).

11. Find the area of the parallelogram with adjacent vectors 3i – j + 2k and 5i + j – k.

12. A triangle has vertices (-1, 2, 4), (3, 7, 5) and (4, 2, 3). Find the area of this triangle.

13. A plane contains the vectors b = 2i – j – k and c = 3i + j + 2k.

a. Find the vector equation of the plane, containing the vectors b and c and passing the point (2, -2, 3)

b. Find the Cartesian equation for part a

c. Express b x c in the form of ai + bj + ck

14. Find an equation of the plane containing the two lines [pic]and [pic].

15. Find the line of intersection of the planes [pic]and [pic].

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