Question

How do I determine the vector projection of a vector?

Answer 1

Hint : Here in the above question we are asked to determine the vector projection of the vector. So firstly we need to know the formula of vector projection and will write down the formula of unit vector which is vector projection of $\overrightarrow b$ onto $\overrightarrow a$ . Later on we will calculate the dot product and calculate the modulus $\overrightarrow a$ in order for getting the desired result.
Formula:
The formula used here will be vector projection of $\overrightarrow b$ onto $\overrightarrow a$
$pro{j_{\overrightarrow a }}\overrightarrow b = \dfrac{{\overrightarrow a \times \overrightarrow b }}{{{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a$

Complete step by step solution:
According to this formula we need to determine the vector projection of the vector. Here we will consider $a$ as one vector and $b$ as another vector.
So here firstly we will write the vector projection of $\overrightarrow b$ onto $\overrightarrow a$
$pro{j_{\overrightarrow a }}\overrightarrow b = \dfrac{{\overrightarrow a \times \overrightarrow b }}{{{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a$
So here we will use the dot product formula $\overrightarrow a \times \overrightarrow b$
${\text{Projection}} = \left| {\overrightarrow b } \right|\sin \theta \times {\text{ unit vector of }}\overrightarrow a$
The unit vector of $a$ can be written as $\dfrac{{\overrightarrow a }}{{\left| {\overrightarrow a } \right|}}$ . Now we will substitute it in the above equation
$projection = \left| {\overrightarrow b } \right|\sin \theta \times \dfrac{{\overrightarrow a }}{{\left| {\overrightarrow a } \right|}}$
Now later on we will multiply both the numerator and denominator with $\left| {\overrightarrow a } \right|$ . So we will get
$projection = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\sin \theta \times \dfrac{{\overrightarrow a }}{{{{\left| {\overrightarrow a } \right|}^2}}}$
Here $\overrightarrow a \times \overrightarrow b = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\sin \theta$
$pro{j_{\overrightarrow a }}\overrightarrow b = \dfrac{{\overrightarrow a \times \overrightarrow b }}{{{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a$
Hence the vector projection of vector is determined and lastly we will calculate the modulus of $\overrightarrow a$ which comes out to be $\left\| {\overrightarrow a } \right\|$
$pro{j_{\overrightarrow a }}\overrightarrow b = (\overrightarrow a \times \overrightarrow {b)} \dfrac{{\overrightarrow a }}{{{{\left| {\overrightarrow a } \right|}^2}}}$

Note : Here it is very important to have understanding of basic formulas of vectors to solve the above problem. Remember for solving such types of problems we need to keep in mind the formula of projection of one vector to another. We need to be aware about the dot product of two vector formulas. Perform the calculation correctly to get the desired result.