Sheet (1)
(Vectors in 3D) - (Dot Product) - (Cross Product)
[1]
If 𝑎⃑ = < 2, −3, 4 > , 𝑏⃑⃑ = < −1, 2, 5 > and 𝑐⃑ = < 3, 6, −1 > Find;
1) 𝑎⃑ ∙ 𝑐⃑
2) 𝑏⃑⃑ ∙ ( 𝑎⃑ − 𝑐⃑ )
3) (2𝑏⃑⃑ ) ∙ (3𝑐⃑)
4) (𝑐⃑ ∙ 𝑏⃑⃑ ) 𝑎⃑
5) 𝑎⃑ × 𝑐⃑
6) 𝑏⃑⃑ × ( 𝑎⃑ × 𝑐⃑ )
7) ‖𝑎⃑ × 𝑏⃑⃑‖
8) 𝑎⃑ ∙ ( 𝑏⃑⃑ × 𝑐⃑ )
9) ‖𝑎⃑ + 𝑏⃑⃑ + 𝑐⃑‖
𝑎⃑⃑
⃑⃑
𝑏
10) ‖‖𝑎⃑⃑‖‖ + 5 ‖‖𝑏⃑⃑‖‖.
1
[2] Find a vector 𝑏⃑⃑ for which ‖𝑏⃑⃑‖ = 2 that is parallel to 𝑎⃑ = < −6, 3, −2 > but has the opposite direction.
[3] Find a vector 𝑣⃑ = < 𝑥1 , 𝑦1 , 1 > that is orthogonal to both 𝑎⃑ = < 3, 1, −1 > and 𝑏⃑⃑ = < −3, 2, 2 >.
[4] Find two unit vectors orthogonal to both vectors 𝐢 + 𝐣 + 𝐤 and 2𝐢 + 𝐤.
[5] Determine a unit vector 𝑎̂ which makes an angle
vector.
𝜋
4
with 𝐤 and is such that 𝑎̂ + 𝐢 + 𝐣 is also a unit
[6] Given the points 𝐴(1, 0, 2), 𝐵(2, 1, 3) and 𝐶(0, −1, 4).
(a) Calculate the angles of Δ𝐴𝐵𝐶.
(b) Find a normal vector to the plane containing 𝐴, 𝐵 and 𝐶.
(c) The area of Δ𝐴𝐵𝐶.
[7] Use vectors to prove that the three points 𝐴(3, 2, −4), 𝐵(9, 8, −10) and 𝐶(−2, −3, 1) are collinear.
[8]
Find the work done by a constant force with vector representation 𝐹⃑ = 4𝐢 + 3𝐣 + 5 𝐤 moves an
object along a straight line from the point 𝑃1 (3, 1, −2) to the point 𝑃2 (2, 4, 6).
[9] Use vectors to decide whether the triangle with vertices 𝐴(1, −3, −2), 𝐵(2, 0, −4) and 𝐶(6, −2, −5) is
right-angled.
[10] Find the area of the parallelogram with vertices 𝐴(1, 3, 2), 𝐵(4, 5, 0), 𝐶(2, 0, 4) and 𝐷(5, 2, 2).
[11] Find 𝜆 if the volume of the parallelepiped whose edges are represented by −12 𝐢 + 𝜆 𝐤,
3 𝐣 − 𝐤 and 2 𝐢 + 𝐣 − 15 𝐤 is 546.
[12] Use triple Scalar product to determine whether the points 𝐴(1, 0, 1), 𝐵(0 , 4 , 6), 𝐶(3, −1, 2) and
𝐷(6, 2, 8) lie in the same plane.
[13] State whether the following statements are true or false. Explain your answer and if false state a
possible correction
(a) If 𝑎⃑ × 𝑏⃑⃑ = 0 and 𝑎⃑ ∙ 𝑏⃑⃑ = 0 then at least one of the two vectors 𝑎⃑ and 𝑏⃑⃑ is the zero vector.
(b) If 𝑎⃑ ∙ 𝑏⃑⃑ = 𝑎⃑ ∙ 𝑐⃑ and 𝑎⃑ × 𝑏⃑⃑ = 𝑎⃑ × 𝑐⃑ ; 𝑎⃑ ≠ ⃑0⃑ then 𝑏⃑⃑ = 𝑐⃑.
Math 4 (Spring 2017 )