Shortcut Methods and Tricks for Solving Numericals on Vector Operations

Here are some common shortcuts and tricks that can help you solve numericals on vector operations more efficiently:

1. Magnitude of a Vector:

• To find the magnitude (length) of a vector (\overrightarrow{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}), use the formula: $$|\overrightarrow{a}|=\sqrt{a_1^2+a_2^2+a_3^2}$$

2. Direction of a Vector:

• The direction of a vector can be expressed using its components as follows: $$\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\Rightarrow \theta ={\tan }^{-1}\left(\frac{a_2}{a_1}\right)$$ where (\theta) is the angle between the vector and the positive (x)-axis.

3. Angle between Two Vectors:

• To find the angle ((\theta)) between two vectors (\overrightarrow{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) and (\overrightarrow{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}), use the formula: $$\cos \theta =\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}$$ where (\overrightarrow{a} \cdot \overrightarrow{b}) is the dot product of the vectors and (|\overrightarrow{a}|) and (|\overrightarrow{b}|) are their respective magnitudes.

4. Dot Product of Two Vectors:

• The dot product of two vectors (\overrightarrow{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) and (\overrightarrow{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}) is given by: $$\overrightarrow{a}\cdot \overrightarrow{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$$

5. Cross Product of Two Vectors:

• The cross product of two vectors (\overrightarrow{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) and (\overrightarrow{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}) is given by the determinant: $$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$$

6. Unit Vector:

• To find the unit vector in the direction of a vector (\overrightarrow{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}), use the formula: $$\hat{a}=\frac{\overrightarrow{a}}{|\overrightarrow{a}|}=\frac{a_1\hat{i}+a_2\hat{j}+a_3\hat{k}}{\sqrt{a_1^2+a_2^2+a_3^2}}$$

7. Scalar Projection:

• The scalar projection of a vector (\overrightarrow{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}) onto a nonzero vector (\overrightarrow{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) is given by: $$\text{scalar projection of }\overrightarrow{b}\text{ onto }\overrightarrow{a}=\frac{\overrightarrow{b}\cdot \overrightarrow{a}}{|\overrightarrow{a}|}$$

8. Vector Projection:

• The vector projection of a vector (\overrightarrow{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}) onto a nonzero vector (\overrightarrow{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) is given by: $$\text{vector projection of }\overrightarrow{b}\text{ onto }\overrightarrow{a}=(\overrightarrow{b}\cdot \hat{a})\hat{a}$$ where (\hat{a}) is the unit vector in the direction of (\overrightarrow{a}).