**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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- Spring '16
- Alveen Chand
- Dot Product, Vector Motors, ~v