Vectors Test
1.
Find the angle between the vectors v = i + j + 2k and w = 2i + 3j + k. Give your answer in radians.
(Total 6 marks)
𝑣×𝑣 = 1×2 +1×3+2×1
= 7
|𝑣| = √1
|𝑣| = √2
⇒
2
2
+1
+1
2
2
+3
2
= √6
= √14
𝑣×𝑣
7
√21
=
=
|𝑣| × |𝑣| √6 × √14
6
𝑣𝑣𝑣𝑣 = 40.2
2.
2
+2
𝑣
A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector
plane π : x + 3y + 2z − 24 = 0.
and meets the
Find the angle that the ray of light makes with the plane.
(1,3,2) × (4,1, −2)
𝑣𝑣𝑣𝑣 =
=
√(1)2 + 32 + 22 + √42 + 12 + (−2)2
3
√294
(Total 6 marks)
= 0.1749
⇒ 𝜃 ≈ 80𝑣
𝑣ℎ𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑣ℎ𝑣 𝑣𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑣ℎ𝑣 𝑣𝑣𝑣ℎ𝑣 𝑣𝑣𝑣 𝑣𝑣𝑣 𝑣ℎ𝑣 𝑣𝑣𝑣𝑣𝑣 𝑣𝑣 10𝑣
3.
The point A is the foot of the perpendicular from the point (1, 1, 9) to the plane
2x + y – z = 6. Find the coordinates of A.
(Total 6 marks)
A= 5,3,7
4.
Consider the points A (1, 3, –17) and B (6, – 7, 8) which lie on the line l.
(a)
Find an equation of line l, giving the answer in parametric form.
(4)
𝑣𝑣 = (5, −10,25)
𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣: 𝑣 = 1 − 5𝑣, 𝑣 = 3 + 10𝑣, 𝑣 = 17 − 25𝑣
(b)
The point P is on l such that
Find the coordinates of P.
is perpendicular to l.
(3)
(Total 7 marks)
𝑣ℎ𝑣 𝑣𝑣𝑣𝑣 𝑣𝑣 𝑣ℎ𝑣𝑣𝑣𝑣ℎ 𝑣𝑣𝑣𝑣𝑣 (0,14, −10)
𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑣𝑣 𝑣ℎ𝑣 𝑣𝑣𝑣𝑣 𝑣 = −1 + 2𝑣, 𝑣 = 6 − 3𝑣, 𝑣 = 3 + 9𝑣
5.
The plane π contains the line
(a)
=
=
and the point (1, −2, 3).
Show that the equation of π is 6x + 2y – 3z = –7.
(7)
⇒ 𝑣 = 1 + 2𝑣, 𝑣 = 1 + 3𝑣, 𝑣 = 5 + 6𝑣
𝑣 = (1,1,5), 𝑣(1, −2,3)
𝑣→ = 𝑣. 𝑣 =
(b)
Calculate the distance of the plane π from the origin.
(4)
(Total 11 marks)