Vectors and Vector Resolution IV
Vector components

A vector, A inclined at an angle \( \theta \) to the horizontal can be resolved into two perpendicular components. The horizontal (Ay) and the vertical (Ax) The following scenarios in this case are:



Horizontal component, Ax = Acos \( \theta \)
Vertical component, Ay= Asin \( \theta \)



Horizontal component, Ay = Asin \( \theta \)
Vertical component, Ax = Acos \( \theta \)

Note: sin 30 = cos 60 = 0.5
cos 30 = sin 60 = 0.8660


Example 1

A ball is moving at 18m/s in a direction inclined at 60⁰ to the horizontal. The horizontal component of it's velocity is? (JAMB)

Solution

$$ V_x = Acos \theta $$ $$ V_x = 18 cos 60(horizontal) $$ $$ V_x = 18 × 0.5 $$ $$ V_x = 9m/s $$

Example 2

A plane flies with a velocity of 1000km/hr in a direction N60⁰E. Find its effective velocity in the easterly and northerly directions.

Solution

Velocity in the Northerly direction is given by: $$ V_y = Vsin \theta $$ $$ V_y = 1000 × sin30 $$ $$ V_y = 500km/hr $$ Velocity in the easterly direction is given by: $$ V_x = 1000 × cos 30 $$ $$ V_x = 866km/hr $$

Example 3

A nail on a vertical wall is pulled by means of a cord attached to its head. If the cord makes an angle of 60⁰ to the horizontal and it exerts a force of 80N on the nail. Find;

  1. The effective force which tends to pull the nail out of the wall
  2. The force which tends to bend the nail
  3. In what direction will the nail move if both forces are applied?

Solution

  1. Force that tends to pull the nail out of the wall

    The pulling force is applied horizontally to the nail $$ F_x = Fcos \theta $$ $$ F_x = 80cos60 $$ $$ F_x = 40N $$

  2. The force that tends to bend the nail

    The force that bends the nail is applied vertically to the nail $$ F_y = Fsin \theta $$ $$ F_y = 80sin 60 $$ $$ F_y = 69.28N $$

  3. In what direction will the nail move if both forces are applied

    4. The nail will move vertically because the bending force is more than the pulling force

Example 4

A force of 10N is applied to a block of wood as illustrated below. What is the vertical component of the force? (NECO)



Solution

$$Resultant force = 10N $$ $$ \text{Vertical component} = Fsin \theta $$ $$ F_y = 10sin 30 $$ $$ F_y = 5N $$


Vector Components Calculator


Input the values for the parameters to solve using the calculator
Summary