Resolving Parallel, Anti-Parallel and Perpendicular Vectors

CSEC Physics Syllabus - Effective for examinations from May - June 2015

Section A - Mechanics

Vectors

Specific Objective 2.3

calculate the resultant of vectors which are parallel, anti-parallel and perpendicular;

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Resolving Parallel Vectors 

Vectors acting in the same direction are resolved by addition. Their combined effect is called the Resultant Vector, R. 

Example 1:

Find the Resultant Vector, R, acting on the box.

Solution:

Since both Forces are acting on the box in the same direction, their Resultant Vector (Force) is determined as:

R = 34 N + 34 N

∴ R = 68 N East

Example 2:

Find the Resultant Vector, R, acting on the box.

Solution:

Since both Forces are acting on the box in the same direction, their Resultant Vector (Force) is determined as:

R = 14 N + 30 N

∴ R = 44 N North

Resolving Anti-Parallel Vectors

Vectors acting in the opposite direction are resolved by subtraction. The smaller vector is subtracted from the larger vector and the result is in the direction of the larger vector and called the Resultant Vector, R. 

Example 1:

Find the Resultant Vector, R, acting on the ball.

Solution:

Since the Forces acting on the ball are in the opposite direction, the Resultant Vector (Force) is determined as: 

R = 8 N - 4 N

∴ R = 4 N West

Example 2:

Find the Resultant Vector, R, acting on the ball.

Solution:

In this example we have three forces acting on the ball. Notice there are the two forces acting in the same direction (South). These forces are parallel vectors and we need to resolve them first as follows:

Total South Force = 8 N + 10 N 

∴ Total South Force = 18 N

We can now find the Resultant Vector, R acting on the ball by resolving the two anti-parallel forces acting on the ball:

 R = 18 N - 6 N

∴ R = 12 N South

Resolving Perpendicular Vectors

Vectors acting perpendicularly to each other can be resolved using Pythagoras Theorem and Trigonometric Ratios.

Pythagoras Theorem

Pythagoras Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Where the opposite is the side opposite the angle, θ; the adjacent is the side next to the angle, θ and the hypotenuse is the side opposite the right angle of the triangle.

The Resultant Vector, R is drawn from the tail of the first vector to the head of the second vector and the angle of action of the Resultant Vector is taken from the first vector.

Example 1:

Find the Resultant Vector, R for the following perpendicular vectors.

Solution:

We shall draw the Resultant Vector, R from the tail of the first vector to the head of the second vector (similar to scale diagrams). The Resultant Vector is identified as the red vector and calculated using Pythagoras Theorem and the angle of action, θ, is determined from the first vector using a Trigonometric Ratio. 

R is calculated using Pythagoras Theorem: 

θ is calculated using the following Trigonometry Ratio:

Therefore, the Resultant Vector is 12.2 km at an angle of 35 degrees from the 10 km vector.

Example 2:

Find the Resultant Vector, R for the following perpendicular vectors.

Solution:

We shall draw the Resultant Vector, R from the tail of the first vector to the head of the second vector (similar to scale diagrams). The Resultant Vector is identified as the red vector and calculated using Pythagoras Theorem and the angle of action, θ, is determined from the first vector using a Trigonometric Ratio. 

R is calculated using Pythagoras Theorem: 

θ is calculated using the following Trigonometry Ratio:

Therefore, the Resultant Vector is 13.6 N at an angle of 17.2 degrees from the 13 N vector.