Homework 7.1
1
(a)
(b)
(c)
(d)
Which of the following quantities are vectors and which are scalars?
Velocity
Mass
Magnetic field
Temperature
(a)
(b)
(c)
(d)
Find the magnitudes of the following vectors:
(3,4)
(2,1,7)
i + 3j
2i – 3j – 2.5k
(a)
(b)
(c)
(d)
⃗⃗⃗⃗⃗ s for the following points A and B:
Find the vector𝐴𝐵
A(3,7) and B(4,9)
A(2,7,4) and B(1,2,3)
A(x , y) and B(p , q)
A(2x, -y, 3z) and B(x + 1, y - 3, 4z)
2
3
4
⃗⃗⃗⃗⃗ = 𝒂 and 𝑂𝐵
⃗⃗⃗⃗⃗ =
OAB is a triangle, and X and Y are the midpoints of OA and OB respectively. Let 𝑂𝐴
⃗⃗⃗⃗⃗
𝒃. Find 𝑋𝑌 in terms of 𝒂 and 𝒃. Show that XY is parallel to AB.
5
OAB is a triangle, with ⃗⃗⃗⃗⃗
𝑂𝐴 = 𝒂 and ⃗⃗⃗⃗⃗
𝑂𝐵 = 𝒃. X and Y are such that ⃗⃗⃗⃗⃗
𝑂𝑋 = 𝜆𝒂 and ⃗⃗⃗⃗⃗
𝑂𝑌 = 𝜇𝒃. If XY is
parallel to AB show that 𝜆 = 𝜇.
6
⃗⃗⃗⃗⃗ = 𝒂 and 𝑂𝐶
⃗⃗⃗⃗⃗ = 𝒄. Find 𝑂𝐵
⃗⃗⃗⃗⃗ in terms of 𝒂 and 𝒄. Find the position
OABC is a parallelogram, with 𝑂𝐴
vectors of the mid-points of OB and AC in terms of 𝒂 and 𝒄. What can you deduce?
7
A, B, C, D are four points in space, with position vectors 𝒂, 𝒃, 𝒄, 𝒅 respectively. I, J, K, L are the
midpoints of AB, BC, CD and DA respectively. Find the position vectors of I, J, K, and L in terms of
⃗ , ⃗⃗⃗⃗
𝒂, 𝒃, 𝒄, 𝒅. Find the vectors ⃗𝐼𝐽
𝐽𝐾 , ⃗⃗⃗⃗⃗
𝐿𝐾 , ⃗⃗⃗
𝐼𝐿. What can you say about figure IJKL?
8
A, B, C have position vectors 𝒂, 𝒃, 𝒄 respectively. Find the position vector of M, the midpoint of BC.
G lies on AM with AG = 2GM. Show that the position vector of G is 13(𝒂+𝒃+𝒄). Show that G also lies on
the lines from b to the midpoint of AC and from C to the midpoint of AB.
(G is a special point for this triangle. For extra credit find out what this point is called and what it
has to do with gravity.)
9
The position vectors of A and B relative to O are 𝒂 and 𝒃. X is the midpoint of OA and Y lies on OB
with OY = 1/3 OB. XY meet AB at Z. Find the position vector of Z.
10
𝒂 and 𝒃 are two non-parallel vectors. The position vectors of X, Y and Z are 𝒂 + 2𝒃, 𝒂 +
2
𝜆𝒂 + 𝒃. Find the value of 𝜆 if X, Y and Z lie on a straight line.
11
⃗⃗⃗⃗⃗ = 𝒂 and 𝑂𝐵
⃗⃗⃗⃗⃗ = 𝒃. Express, in terms of 𝒂 and 𝒃 the
ABCDEF is a regular hexagon, centre O with 𝑂𝐴
⃗⃗⃗⃗⃗ , 𝐷𝐸
⃗⃗⃗⃗⃗ , 𝐶𝐹
⃗⃗⃗⃗⃗ .
⃗⃗⃗⃗⃗ , 𝐶𝐷
vectors 𝐹𝐴
1
Hints and solutions: http://mathsurgery.wikispaces.com/Phys+AS+unit+2+section+7.1+-+vectors+and+scalars
3
𝒃
4
and