### Shortcut Methods

**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: $$ \overrightarrow{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \quad \Rightarrow \quad \theta = \tan^{-1}\left( \frac{a_2}{a_1} \right)$$ 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: $$\overrightarrow{a} \cdot \overrightarrow{b} = a_1b_1 + a_2b_2 + a_3b_3$$

**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: $$\overrightarrow{a} \times \overrightarrow{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$

**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}).