Question

Resolve the vector into components. 1. A vector starting at the point Q = (4, 6) and ending at the point P = (1, 2).

2. A vector starting at the point P = (1, 2) and ending at the point Q = (4, 6).

Answer 1

1. Horizontal component = -3 units

Vertical component = -4 units

2. Horizontal component = 3 units

Vertical component = 4 units

Explanation:

If a vector M, starting at point A = (a, b) and ending at point B = (c, d) is given, then the vector can be resolved into x and y components as follows;

M = AB

Where;

AB = B - A

AB = (a, b) - (c, d)

AB = (a-c, b-d)

AB = (a-c)i + (b-d)k

Therefore, a-c and b-d are the x and y components of the vector M.

(1) Let the vector be M:

Starting point of M = Q = (4, 6)

Ending point of M = P = (1, 2)

So,

M = PQ

Where;

PQ = Q - P

PQ = (1,2) - (4,6)

PQ = (1-4, 2-6)

PQ = (-3, -4)

Therefore,

M = PQ = -3i - 4j

The x and y components of M are therefore, -3 and -4 respectively.

(2) Let the vector be M:

Starting point of M = P = (1, 2)

Ending point of M = Q = (4, 6)

So,

M = PQ

Where;

PQ = Q - P

PQ = (4,6) - (1,2)

PQ = (4-1, 6-2)

PQ = (3, 4)

Therefore,

M = PQ = 3i + 4j

The x and y components of M are therefore, 3 and 4 respectively.

Note: The x and y components are also called the horizontal and vertical components respectively.