MTH405_Wk_7_Tutorial - MTH405 Wk 7 Tutorial 1 Sketch the...

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Unformatted text preview: MTH405 Wk 7 Tutorial 1. Sketch the vectors with their initial points at the origin. b.) − 5ˆi + 3ˆj d.) 3ˆi − 2ˆj a.) h2, 5i c.) h−5, −4i −−−→ 2. Find the components of the vector P1 P2 and sketch them. (Note that −−−→ vectors denoted as P1 P2 have starting point as P1 and terminating point as P2 .) (a) P (3, 5), P (2, 8) (b) P1 (0, 0), P2 (3, 4) (c) P1 (4, 1), P2 (0, 0) 3. Let p~ = 3ˆi − kˆ, ~q = ˆi − ˆj + 2kˆ and ~r = 3ˆj . Find a.) ~r − ~q b.) 6~ p + 4~r c.) − ~q − 2~r d.) 4(3~ p + ~q) e.) − 8(~q + ~r) + 2~ p f.) 3~r − (~q − ~r) 4. For the following vectors, nd its norm and then use it to normalise the vectors. a.) h3, 4i c.) h0, 3, 0i √ √ b.) 2ˆi − 7ˆj d.) ~v = ˆi + ˆj + kˆ 5. Let ~u = ˆi − 3ˆj + 2kˆ, ~v = ˆi + ˆj , and w ~ = 2ˆi + 2ˆj − 4kˆ. Find a.) ||~u|| ~ c.) ||w|| e.) ||3~u + 5~v + w|| ~ 1 g.) w ~ ||w|| ~ b.) ||~v || d.) ||~u + ~v || 1 f.) w ~ ||w|| ~ 1 6. Find the unit vectors that satisfy the stated conditions. (a) Same direction as −ˆi + 4ˆj . (b) Oppositely directed to 6ˆi − 4ˆj + 2kˆ. (c) Same direction as the vector from the point A(−1, 0, 2) to the point B(3, 1, 1). 7. In each part, nd the dot product of the vectors and the angle between them. (a) ~u = ˆi + 2ˆj , ~v = 6ˆi − 8ˆj . (b) ~u = h−7, −3i, ~v = h0, 1i. (c) ~u = ˆi − 3ˆj + 7kˆ, ~v = 8ˆi − 2ˆj − 2kˆ. (d) ~u = h−3, 1, 2i, ~v = h4, 2, −5i. 8. In each part, use the information given to nd ~u · ~v . (a) ||~u|| = 1, ||~v || = 2, the angle between ~u and ~v is π6 . (b) ||~u|| = 2, k|~v || = 3, the angle between ~u and ~v is 135◦ . 9. In each part, nd the vector component of ~v along ~b and the vector component of ~v orthogonal to ~b. Then sketch the vectors ~v , projb~v and ~v − projb~v . (a) ~v = 2ˆi − ˆj , ~b = 3ˆi + 4ˆj . (b) ~v = h4, 5i, ~b = h1, 2i. (c) ~v = −3ˆi − 2ˆj , ~b = 2ˆi + ˆj . 10. Show that the three vectors ~v1 ~v2 ~v3 3ˆi − ˆj + 2kˆ = ˆi + ˆj − kˆ = ˆi − 5ˆj − 4kˆ = are pairwise orthogonal. 2...
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