Projection & Projection Vector

In the study of vectors, the concepts of projection and projection vector are crucial for understanding how vectors interact with each other in terms of their components along specific directions. Let's delve into these concepts in detail.

Projection

The projection of a vector onto another vector is a way to express how much of one vector lies in the direction of another vector. It is a scalar quantity that represents the component of one vector along the direction of another vector.

Formula for Projection

If we have two vectors $\vec{A}$ and $\vec{B}$, the projection of $\vec{A}$ onto $\vec{B}$, denoted as $\text{proj}_{\vec{B}}\vec{A}$, is given by the formula:

$$ \text{proj}_{\vec{B}}\vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \vec{B} $$

Here, $\vec{A} \cdot \vec{B}$ represents the dot product of vectors $\vec{A}$ and $\vec{B}$, and $|\vec{B}|$ is the magnitude of vector $\vec{B}$.

Projection Vector

The projection vector is the vector representation of the projection of one vector onto another. It is a vector that lies in the direction of the second vector and has a magnitude equal to the scalar projection.

Formula for Projection Vector

The projection vector of $\vec{A}$ onto $\vec{B}$, denoted as $\vec{proj}_{\vec{B}}\vec{A}$, is given by the formula:

$$ \vec{proj}_{\vec{B}}\vec{A} = \left( \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \right) \vec{B} $$

This formula is similar to the scalar projection, but it gives us a vector quantity.

Differences and Important Points

Here is a table summarizing the differences and important points between projection and projection vector:

Aspect	Projection (Scalar)	Projection Vector (Vector)
Nature	Scalar quantity	Vector quantity
Formula	$\frac{\vec{A} \cdot \vec{B}}{\	\vec{B}\
Direction	Not applicable	In the direction of $\vec{B}$
Magnitude	Length of the component of $\vec{A}$ along $\vec{B}$	Same as the scalar projection
Representation	Numerical value	Vector with both magnitude and direction

Examples

Let's go through some examples to clarify these concepts.

Example 1: Scalar Projection

Given two vectors $\vec{A} = 3\hat{i} + 4\hat{j}$ and $\vec{B} = 2\hat{i} + 0\hat{j}$, find the scalar projection of $\vec{A}$ onto $\vec{B}$.

First, we calculate the dot product of $\vec{A}$ and $\vec{B}$:

$$ \vec{A} \cdot \vec{B} = (3\hat{i} + 4\hat{j}) \cdot (2\hat{i} + 0\hat{j}) = 3 \times 2 + 4 \times 0 = 6 $$

Next, we find the magnitude of $\vec{B}$:

$$ |\vec{B}| = \sqrt{2^2 + 0^2} = \sqrt{4} = 2 $$

Now, we can calculate the scalar projection:

$$ \text{proj}_{\vec{B}}\vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} = \frac{6}{2} = 3 $$

Example 2: Projection Vector

Using the same vectors from Example 1, let's find the projection vector of $\vec{A}$ onto $\vec{B}$.

We already have the dot product $\vec{A} \cdot \vec{B} = 6$ and the magnitude $|\vec{B}| = 2$. The projection vector is given by:

$$ \vec{proj}_{\vec{B}}\vec{A} = \left( \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \right) \vec{B} = \left( \frac{6}{2^2} \right) (2\hat{i} + 0\hat{j}) = \left( \frac{6}{4} \right) 2\hat{i} = 3\hat{i} $$

The projection vector $\vec{proj}_{\vec{B}}\vec{A}$ is $3\hat{i}$, which lies in the direction of $\vec{B}$ and has a magnitude of 3, the same as the scalar projection.

Understanding projection and projection vector is essential for various applications in physics and engineering, such as resolving forces into components, analyzing motion, and performing vector operations in three-dimensional space.