We know that the vector is a quantity that has both magnitude and direction. There are different types of vectors, such as unit vector, zero vector, collinear vector, equal vector, and so on. The scalar components of a vector are its direction ratios and represent the scalar projections along their respective axes. In this article, we are going to discuss the projection of a vector on a line with many solved examples.

What is Meant by Projection of a Vector on a Line?

Assume that the vector AB makes an angle θ with the directed line, say “l” in the anti-clock direction as shown in the figure.

Thus, the projection of vector AB on the directed line “l” is a vector

Here, the vector

In the above figure (i) and (iv), represents the projection of the vector AB along the directed line “l” is the vector AC.

Note:

• If the unit vector
  $\begin{array}{l}\hat{p}\end{array}$
  is along the directed line l, then the projection of
  $\begin{array}{l}\overrightarrow{a}\end{array}$
  on the directed line l is given by
  $\begin{array}{l}\overrightarrow{a}.\hat{p}\end{array}$
  .
• The projection of vector “a” on the vector “b” is given by:
  $\begin{array}{l}\overrightarrow{a}.\hat{b}\end{array}$
  or
  $\begin{array}{l}\overrightarrow{a}.\left(\frac{\overrightarrow{b}}{|\overrightarrow{b}|}\right)\end{array}$
  or
  $\begin{array}{l}\frac{1}{|\overrightarrow{b}|}(\overrightarrow{a}.\overrightarrow{b})\end{array}$
• If θ = 0, then the projection vector of
  $\begin{array}{l}\overline{AB}\end{array}$
  will be
  $\begin{array}{l}\overline{AB}\end{array}$
  .
• If θ = π, then the projection of
  $\begin{array}{l}\overline{AB}\end{array}$
  will be
  $\begin{array}{l}\overline{BA}\end{array}$
  .
• If θ = π/2 or θ = 3π/2, then the projection of
  $\begin{array}{l}\overline{AB}\end{array}$
  will be a zero vector.

If α, β and γ be the direction angles of the vector

It is observed that

In case

Projection of a Vector on a Line Examples

Example 1:

Determine the projection of the vector

Solution:

Given:

We know that the formula to find the projection of

= [2(1) + 3(2) + 2(1)]/√[(1)²+(2)²+(1)³]

= 10/√6

= (5/3)√6.

Hence, the projection of vector

Example 2:

Find

Solution:

Given that:

Also, given that

Now, simplify the above equation,

Therefore,

As, the magnitude of a vector cannot be negative, the value of

Example 3:

Find the angle between two vectors

Solution:

Given:

The formula to find the angle between two vectors is given by:

Hence,

= 1-1-1

= -1

Therefore,

Now, substituting the value in the formula, we get

Cos θ = -⅓

Hence, the angle between two vectors is θ = cos^-1(-⅓).

Practice Problems

Solve the following problems:

1. Determine the projection of the vector
   $\begin{array}{l}\hat{i}–\hat{j}\end{array}$
   on the vector
   $\begin{array}{l}\hat{i}+\hat{j}\end{array}$
   .
2. Determine the projection of vector
   $\begin{array}{l}\hat{i}+3\hat{j}+7\hat{k}\end{array}$
   on the vector
   $\begin{array}{l}7\hat{i}–\hat{j}+8\hat{k}\end{array}$
3. Compute the angle between two vectors
   $\begin{array}{l}\hat{i}–2\hat{j}+3\hat{k}\end{array}$
   and
   $\begin{array}{l}3\hat{i}–2\hat{j}+\hat{k}\end{array}$
   .

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