1. **Problem Statement:** Understand the fundamental concepts and properties of vectors in three-dimensional space $\mathbb{R}^3$ including unit vectors, components, magnitude, equality, zero vector, negative vector, scalar multiplication, parallel vectors, vector addition, and properties of scalar multiplication. 2. **Unit Vectors:** The unit vectors along the coordinate axes are $\vec{i}$, $\vec{j}$, and $\vec{k}$ along the $x$-, $y$-, and $z$-axes respectively. 3. **Components of a Vector:** A vector $\vec{r} = \overrightarrow{OP}$ for point $P(x,y,z)$ can be expressed as: $$\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$$ where $x$, $y$, and $z$ are the components along the $x$-, $y$-, and $z$-axes. 4. **Magnitude of a Vector:** The magnitude $|\vec{r}|$ is given by: $$|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$$ This follows from the Pythagorean theorem in 3D space. 5. **Unit Vector in Direction of $\vec{r}$:** The unit vector $\hat{r}$ is: $$\hat{r} = \frac{\vec{r}}{|\vec{r}|} = \frac{x}{\sqrt{x^2 + y^2 + z^2}}\vec{i} + \frac{y}{\sqrt{x^2 + y^2 + z^2}}\vec{j} + \frac{z}{\sqrt{x^2 + y^2 + z^2}}\vec{k}$$ 6. **Equal Vectors:** Two vectors $\vec{r}_1 = x_1\vec{i} + y_1\vec{j} + z_1\vec{k}$ and $\vec{r}_2 = x_2\vec{i} + y_2\vec{j} + z_2\vec{k}$ are equal if and only if: $$x_1 = x_2, \quad y_1 = y_2, \quad z_1 = z_2$$ 7. **Zero Vector:** The zero vector $\vec{0}$ has all components zero: $$\vec{0} = 0\vec{i} + 0\vec{j} + 0\vec{k}$$ 8. **Negative of a Vector:** For $\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$, the negative is: $$-\vec{r} = (-x)\vec{i} + (-y)\vec{j} + (-z)\vec{k}$$ 9. **Scalar Multiplication:** For scalar $\lambda$ and vector $\vec{r}$: $$\lambda \vec{r} = (\lambda x)\vec{i} + (\lambda y)\vec{j} + (\lambda z)\vec{k}$$ 10. **Parallel Vectors:** Two non-zero vectors $\vec{r}_1$ and $\vec{r}_2$ are parallel if there exists non-zero scalar $\lambda$ such that: $$\vec{r}_1 = \lambda \vec{r}_2$$ which implies: $$\frac{x_1}{x_2} = \frac{y_1}{y_2} = \frac{z_1}{z_2} = \lambda$$ 11. **Addition of Two Vectors:** For $\vec{r}_1 = x_1\vec{i} + y_1\vec{j} + z_1\vec{k}$ and $\vec{r}_2 = x_2\vec{i} + y_2\vec{j} + z_2\vec{k}$: $$\vec{r}_1 + \vec{r}_2 = (x_1 + x_2)\vec{i} + (y_1 + y_2)\vec{j} + (z_1 + z_2)\vec{k}$$ 12. **Properties of Vector Addition:** - Commutative Law: $\vec{r}_1 + \vec{r}_2 = \vec{r}_2 + \vec{r}_1$ - Associative Law: $\vec{r}_1 + (\vec{r}_2 + \vec{r}_3) = (\vec{r}_1 + \vec{r}_2) + \vec{r}_3$ - Identity Element: $\vec{0} + \vec{r} = \vec{r} + \vec{0} = \vec{r}$ - Additive Inverse: $\vec{r} + (-\vec{r}) = (-\vec{r}) + \vec{r} = \vec{0}$ 13. **Properties of Scalar Multiplication:** - Commutative Law: $\lambda \vec{r} = \vec{r} \lambda$ - Associative Law: $\lambda_1 (\lambda_2 \vec{r}) = (\lambda_1 \lambda_2) \vec{r}$ - Distributive Laws: (i) $(\lambda_1 + \lambda_2) \vec{r}_1 = \lambda_1 \vec{r}_1 + \lambda_2 \vec{r}_1$ (ii) $\lambda_1 (\vec{r}_1 + \vec{r}_2) = \lambda_1 \vec{r}_1 + \lambda_1 \vec{r}_2$ This completes the fundamental definitions and properties of vectors in $\mathbb{R}^3$.