MATH II SEF1124
SEMESTER III, 2011/12
EXTRA EX: CHAPTER 3
POLAR COORDINATES
1.
(a)
The rectangular coordinates of the point P is (-5, 12). Find a polar coordinates (r, 𝜃) for point P
with r > 0 and -3600< 𝜃< 00.
(b)
(13, -247.380)
The rectangular coordinates of the point P is (7, 24). Find a polar coordinates (r, 𝜃) for point P
with r<0 and 00< 𝜃<3600
2.
(-25 , 253.740)
Find the polar coordinates of a point whose rectangular coordinates is (3√3, −3)
with r < 0 and −2𝜋 < 𝜃 < 0
Then plot the point.
(−6, −
7𝜋
6
)
3.
Express the equation 𝑥(𝑥 2 + 𝑦 2 ) = 4 in polar form.
4.
Convert the polar equation 𝑟 = −2𝑐𝑜𝑠𝜃 to rectangular form.
5.
Transform the polar equation 𝑟 + 3 sin 𝜃 = 2𝑐𝑠𝑐𝜃 to rectangular form.
6.
Given the complex numbers z and w. Find zw and
𝑧 = 4 (𝑐𝑜𝑠
5𝜋
8
+ 𝑖 𝑠𝑖𝑛
5𝜋
8
) and w = 2 (𝑐𝑜𝑠
7.
Write (−√3 − 𝑖)6 in the standard form 𝑎 + 𝑏𝑖.
8.
If 𝑧 = 3 (𝑐𝑜𝑠
𝜋
5
Find 𝑧 3 , 𝑤 −2 and
9.
𝜋
3𝜋
5
5
+ 𝑖 𝑠𝑖𝑛 ) and 𝑤 = (𝑐𝑜𝑠
𝑧3
𝑤2
+ 𝑖 𝑠𝑖𝑛
𝑧
𝐰
9𝜋
16
3𝜋
5
. Leave your answer in polar form.
+ 𝑖 𝑠𝑖𝑛
9𝜋
16
)
)
.
Given the complex number 𝑧 = −√3 − 𝑖.
(a) Express z in polar form giving the argument in degrees.
(b) Find 𝑧 4 .
(c) Find the complex cube roots of z.
10. Find all the fourth roots of −5 + 5√3𝑖. Write all answers in the form 𝑎 + 𝑏𝑖 and then plot them in
rectangular coordinates.
VECTORS
1.
Using vectors, find the distance between P(1, 5) and Q(-2, -6).
2.
Given v = -2i -3j and w= i + j , find
(a) v - w
(b) |𝒗 − 𝒘|
Hence find a unit vector in the direction v – w.
3.
Find a vector of magnitude 5 units and parallel to vector p = 6i + 8j.
4.
Given vectors a = i + j +k,
5.
b = 2i – 2j and c = -i – 2j + 5k
i)
Find the vector 2a + b and show that 2a + b is parallel to the vector 2i + k
ii)
Show that |𝒂 + 𝒃 + 𝒄| = 7.
Given vectors p = j + 2k
q = i – j + mk ( m is a constant) determine the vector p – 2q and
hence find the possible values of m if |𝒑 − 2𝒒| = 7 units.
MATH II SEF1124
6.
SEMESTER III, 2011/12
EXTRA EX: CHAPTER 3
Position vectors of A, B, C and D relative to the origin are i – j, 2i +j + k, 4i + 2j + 3k and
3i
+ k respectively. Find vectors
(a)
7.
⃗⃗⃗⃗⃗
𝐴𝐷
(b)
⃗⃗⃗⃗⃗
𝐵𝐶
Given that a = 3i + 2j + k
and b = -2i + mj – 2k where m is a constant.
Express a + 3b in terms of i, j and k. Find the possible values of m if |𝒂 + 3𝒃| = 5 √2.
8.
Given p = -i -2j + 6k , q = 3i + 4j + 5k and r = -2i + j –k
a) Show that p + q + r is parallel to 2i + ½ j + 6k
b) Find the value of |𝑝 − 𝑞 + 𝑟|
9.
Vectors p, q and r are 4i + 2j – 3k, 6i + 5j – k and 2i + 3j + 2k respectively. Show that
vectors p, q and r form a triangle.
10. If m = 7i + 4j -3k, n = -6i – 3j + 4k and p = i -9j + 10k, find
a) The angle between the vectors m and n
b) 𝒎 ∙ (𝒏 − 𝒑)
c) (𝒎 − 𝒏) ∙ (𝒎 + 𝒏)
11. If
a)
b)
c)
p = 4i + 2j – k , q = 6i + 6j – 2k and r = -3i – 4j – 3k,
(𝒒 ∙ (𝒑 − 2𝒓)
(𝒓 + 𝒒) ∙ 𝒑)
(𝒑 − 𝒒) ∙ 𝒓
evaluate
12. Show that p = 2i +9j + 5k and q = -3i + 4j – 6k are perpendicular to each other.
A vector r = 𝜆2 𝒊 − (𝜆 + 4)𝒋 +
10
11
𝒌
is perpendicular to p – q.
Find the values of λ.
13. Find the angle, in degrees, between vectors p = 2i + 9j – 4k and
correct to one decimal place.
q = -3i +5j + 6k
14. If α, β and γ are the direction angles of ⃗⃗⃗⃗⃗
𝑂𝑃 = 2i – 3j – 3k , find the direction cosines of ⃗⃗⃗⃗⃗⃗
𝑂𝑃.
15. Given α, β and γ are the direction angles of ⃗⃗⃗⃗⃗
𝑂𝑃 = -5i + 2j + 3k , find the values of α, β and γ .
16. A vector makes an inclination of 45°, θ and 135° with the positive x, y and z respectively.
Find the values of cos θ and hence , the vector if
|𝒓| = 4.
17. Find a vector perpendicular to both vectors v = 2i + j + 3k and w = 4i – 4j + 6k.
18. Find a unit vector orthogonal to both vectors p = i + 2j – k and q = 2i – j + 3k. Hence, find a
vector with magnitude √27 in the direction of p x q.
19. Find a vector of magnitude 6 which is perpendicular to both v = 4i – j + 2k and
20. If a = 5i + mj – 2k and b = 10i + 6j + 4k, find
a) the value of m if a and b are orthogonal to each other
b) a vector orthogonal to both a and b .
w = 3i + 2j – k.