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The magnitude of the vector (3, 4) is 5 and the dot product of (1, 2, 2) with (3, −1, 2) is 5, so this IB sheet opens with a magnitude and one scalar product of two vectors. The answers record (3/5, 4/5) and (3, 3).
Key Takeaways
The magnitude of (3, 4) is 5 (Q1).
(1, 2, 2) · (3, −1, 2) = 5 (Q2).
Q4 records (3, 3).
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.
What is the magnitude of the vector (3, 4)?
(a) 5 (b) 7 (c) 12 (d) 1
2.
What is the dot product of (1, 2, 2) and (3, −1, 2)?
(a) 3 (b) 5 (c) −5 (d) 6
3.
What is the unit vector in the direction of (3, 4)?
(a) (3/5, 4/5) (b) (3, 4) (c) (4/5, 3/5) (d) (3/7, 4/7)
4.
If a = (2, 1) and b = (1, 2), what is a + b?
(a) (3, 3) (b) (2, 2) (c) (1, 1) (d) (4, 3)
5.
What is the dot product of two perpendicular vectors?
(a) Their product (b) 0 (c) 1 (d) Their sum
6.
What is the distance between the points (1, 2) and (4, 6)?
(a) 5 (b) 25 (c) 7 (d) 3
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.
Find the magnitude of v = (−3, 4).
8.
If a = (2, −1, 3) and b = (1, 4, −2), find a + b.
9.
Find the dot product of (4, −1, 2) and (2, 3, −1).
10.
Find the unit vector in the direction of (0, 5).
11.
Are the vectors (1, 2) and (2, −1) perpendicular?
12.
Find the position vector of the midpoint of A(1, 2) and B(5, 6).
13.
Find k such that (k, 4) has magnitude 5, given that k is positive.
14.
If a = (2, 3) and b = (4, 5), find 2a − b.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.
A boat travels 30 km east then 40 km north. Find the boat's displacement from its starting point.
16.
Forces F1 = (3, 4) N and F2 = (5, −2) N act on a body. Find the resultant force.
17.
A plane flies with velocity (200, 150) km/h for 2 hours from the origin. Find its distance from the origin.
18.
Find the angle between the vectors (1, 0, 0) and (0, 1, 0).
19.
A particle at position (2, 3) moves along the vector (4, −1). Find its new position.
20.
Show that the vectors a = (1, 2, 2) and b = (2, 1, −2) are perpendicular.
Answer Key
1. a) 5
2. b) 5
3. a) (3/5, 4/5)
4. a) (3, 3)
5. b) 0
6. a) 5
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is the magnitude of the vector (3, 4)?
5, option (a) in this worksheet (Q1).
What is the dot product of (1, 2, 2) and (3, −1, 2)?
5, option (b) in this worksheet (Q2).
What unit vector is recorded in the direction of (3, 4)?
(3/5, 4/5), option (a) in this worksheet (Q3).