Grade 10 core mathematics – Vectors I Quiz
1. Which of the following best describes a vector?
A vector is defined by having both magnitude (size) and direction, unlike scalars which have magnitude only.
2. How is the vector from point A to point B commonly written?
The standard notation for the vector starting at A and ending at B is written as vector AB (often shown with an arrow AB).
3. What is the magnitude (length) of the vector (3, 4)?
Magnitude = sqrt(3^2 + 4^2) = sqrt(9+16) = sqrt(25) = 5.
4. Which is the unit vector in the direction of (3, 4)?
The unit vector is the vector divided by its magnitude: (3,4)/5 = (3/5,4/5).
5. Two vectors are equal if they have the same...
Equal vectors must have identical magnitude and direction; their positions (starting points) may differ.
6. Find the resultant of the vectors (2, 1) and (1, 3).
Vector addition is component-wise: (2+1, 1+3) = (3,4).
7. What is 3 times the vector (2, -1)?
Scalar multiplication: 3*(2, -1) = (3*2, 3*(-1)) = (6, -3).
8. What is the vector from P(1, 2) to Q(4, 6)?
Vector PQ = Q - P = (4-1, 6-2) = (3,4).
9. What is the position vector of point (3, -2) relative to the origin?
The position vector from the origin to (3,-2) is simply (3,-2).
10. What is the magnitude of the vector from (-1, 0) to (2, -4)?
Vector = (2-(-1), -4-0) = (3,-4); magnitude = sqrt(3^2+(-4)^2)=sqrt(9+16)=5.
11. What is the midpoint of A(1, 3) and B(5, 7)?
Midpoint = ((1+5)/2, (3+7)/2) = (3,5).
12. In parallelogram ABCD, AB = (2, 0) and AD = (0, 3). What is AC?
Diagonal AC = AB + AD = (2,0)+(0,3) = (2,3).
13. What is the opposite of the vector (4, -2)?
The opposite (negative) of (4,-2) is (-4,2), reversing both components.
14. Are the vectors (2, 4) and (3, 6) collinear?
(3,6) = 1.5*(2,4), so one is a scalar multiple of the other and they lie on the same line (are collinear).
15. Which pair of vectors is perpendicular?
Dot product = 1*2 + 2*(-1) = 2 - 2 = 0; zero dot product means perpendicular.
16. What is the length of the unit vector (5/13, 12/13)?
Unit vectors have length 1; check: sqrt((5/13)^2+(12/13)^2)=sqrt(25/169+144/169)=sqrt(169/169)=1.
17. If a vector v has magnitude 10 and is in the direction of (3, 4), what is v?
Unit vector in that direction is (3/5,4/5); multiply by 10 gives v = 10*(3/5,4/5) = (6,8).
18. If AB = (2, 1) and BC = (1, 2), what is AC?
By the triangle rule AC = AB + BC = (2+1, 1+2) = (3,3).
19. If OA = (2, 3) and OB = (5, 7) (position vectors from the origin), what is AB?
AB = OB - OA = (5-2, 7-3) = (3,4).
20. If |u| = 3, what is |2u|?
Scaling a vector by 2 scales its magnitude by 2: |2u| = 2|u| = 2*3 = 6.
21. Which of the following is a unit vector perpendicular to (1, 0)?
(0,1) has length 1 and dot product with (1,0) is 0, so it is a unit vector perpendicular to (1,0).
22. If vector v = (x, 4) has magnitude 5 and x is positive, what is x?
Solve sqrt(x^2+4^2)=5 → x^2+16=25 → x^2=9 → positive x = 3.
23. Find the value of k so that (k, 2) is perpendicular to (3, -6).
Perpendicular when dot product is zero: 3k + 2*(-6) = 0 → 3k -12 =0 → k = 4.
24. What is u - v if u = (2, 1) and v = (1, -1)?
Subtract components: (2-1, 1-(-1)) = (1, 2).
25. If u + v = (5, 2) and u = (2, 3), what is v?
v = (5,2) - u = (5-2, 2-3) = (3, -1).