# How to Calculate Scalar and Vector Projections

## How to Find the Vector Projection

The formula for the vector projection of a onto b is equal to [ab] / [bb] (b).

The formula for the vector projection of onto

In this formula:

• is pronounced as ‘the projection of vector a onto the vector b
• Each vector is made up of and in 2D or and in 3D.
• is the dot product, calculated by in 2D or in 3D.
• Similarly, in 2D and in 3D
To calculate the vector projection of ‘a’ onto ‘b’:

1. Calculate the dot product of ‘a’ and ‘b’.
2. Divide this by the dot product of ‘b’ and ‘b’.
3. Multiply this by the vector ‘b’.

For example, calculate the vector projection of on .

We will use the formula, , where and .

We first find the dot product of and .

The dot product is calculated using .

Therefore, .

and .

Therefore, the formula of becomes .

Since , the vector projection is

For example, calculate the vector projection of the vector on .

The formula is used with and ,

.

.

Therefore, the formula of becomes .

Since , or .

### How to Calculate a Vector Projection in 3D

Here is an example of calculating a vector projection in 3 dimensions.

The formula for calculating a vector projection in 3D is still .

Calculate the projection of on .

Here, and .

The dot product of these vectors is calculated by multiplying the 𝑥 components, y components and z components together separately.

Therefore, the vector projection equation, becomes .

This can be simplified to or .

## What is Vector Projection

The vector projection of vector ‘a’ on vector ‘b’ tells us what component of vector ‘a’ acts in the direction of vector ‘b’. The result is a vector. If the vector projection is negative, vector ‘a’ is acting in the opposite direction to vector ‘b’.

A simple way to explain vector projection is to consider the shadow formed when a light is shined at right-angles to the vector being projected on.

1. Consider two vectors ‘a’ and ‘b’.
2. Consider a light shining perpendicular to vector ‘b’ from above vector ‘a’.
3. The vector projection of ‘a’ onto ‘b’ is the shadow formed on ‘b’ from the result of this light.

The vector projection notation for the projection of vector ‘a’ on vector ‘b’ is projb’ ‘a’.

Similarly the projection vector of vector ‘b’ on vector ‘a’ is proja’ ‘b’.

Vector projections are useful in real life applications to better understand how forces applied in different directions can impact motion. For example, the effects of windspeed on aeroplanes or the effect of currents on a boat.

A real life example of vector projection is the case of a box being pulled up a slope by a rope that is inclined at an angle.

In this example, the projection vector of the rope onto the slope tells us how much of this force is used to pull the box in the direction of the slope.

## Vector Projection vs Scalar Projection

The vector projection describes the components of a vector that act in the direction of another given vector whereas the scalar projection is the magnitude or length of this vector. The scalar projection is the magnitude of the vector projection.

The vector projection is a vector and provides information about each component of the projection.

The scalar projection is a scalar and simply provides the overall magnitude of the vector but does not tell us the direction it is acting in.

The vector projection can be positive, negative or zero.

• A positive vector projection means that the vectors are acting in the same direction.
• A negative vector projection means that the vectors are acting in the opposite direction.
• A vector projection equal to zero means that the vectors are at right angles.

In the example of the box above, going up the slope is taken in the positive direction. Therefore when the rope is pulling the box up the slope, the rope is acting in the same direction as the slope and the projection vector is positive. When the rope is pulling the box down the slope, the rope is acting in an opposite direction to the slope and the projection vector is negative.

When the rope is perpendicular to the slope, the rope does not pull the block up or down the slope. Therefore, the projection vector is zero.

The scalar projection is simply the magnitude of the force of the vector. Therefore if the rope is pulled with the same strength in each of the three cases, the scalar projection will be equal in each of these cases.

The scalar projection is the magnitude of the vector projection. To calculate the scalar projection, square the components of the vector projection, add them and then square root. For example, if the vector projection is 3i + 4j, then the scalar projection is √(32 + 42) = 5.

## How to Calculate the Scalar Projection

The scalar projection of ‘a’ on ‘b’ is found using |ab| ÷ |b|, where |ab| = a𝑥b𝑥 + ayby and |b| = √(b𝑥2 + by2). For example, the scalar projection of (2, 1) on (3, 4) is (2×3 + 1×4) ÷ √(32 + 42) = 2.

The formula for calculating the scalar projection

For example, calculate the scalar projection of the vector on the vector .

Since is projected onto , is vector and is vector .

The scalar projection equation, becomes .

.

Therefore the scalar projection is .

### Scalar Projection: Real Life Example

A box is pulled up a slope by a rope.

The direction vector of the slope is and the rope is pulled with a force vector of N.

Calculate the magnitude of the force acting in the direction of the slope.

Since we want the magnitude of the force, the scalar projection is required.

In this example, the force is projected in the direction of the slope.

We choose the vector to be the vector that is being projected and the vector to be the vector that it is projected on to.

Therefore and .

The equation of the scalar projection, becomes .

The dot product, .

The magnitude of vector , .

Therefore the scalar product, becomes .

Therefore 8.43 N of force from the rope acts in the direction of the slope.

## Vector Projection Calculator

This vector projection calculator calculates the projection of the vector A onto the vector B.

To use the calculator, simply input the 𝑥, y and z components of both vectors.

In this calculator, Vector A is always the vector being projected onto Vector B so it is important to put the vectors in the correct order.

## Vector Projection Proof

The scalar projection of ‘a’ on ‘b’ is given by a cosθ, where θ is the angle between the vectors. Substituting cosθ = (ab) / ( |a||b| ) into this, we obtain a (ab) / ( |a| |b| ). The vector projection is found by multiplying this by the unit vector in the direction of b to obtain (ab) b / (bb).

1. The scalar projection of on is given by
2. Substitute into this expression to obtain the scalar projection as . This simplifies to .
3. The vector projection is found by multiplying this scalar projection by the unit vector in the direction of the vector : which simplifies to .