1. **Problem Statement:** We are given fundamental definitions related to vectors in space, including unit vectors, equal vectors, zero vectors, negative vectors, scalar multiplication, and parallel vectors. We will explain each concept clearly with formulas and examples. 2. **Unit Vector:** A unit vector \(\hat{r}\) in the direction of a vector \(\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}\) is defined as the vector divided by its magnitude: $$\hat{r} = \frac{\vec{r}}{|\vec{r}|}$$ where the magnitude \(|\vec{r}| = \sqrt{x^{2} + y^{2} + z^{2}}\). Expanding this, we get: $$\hat{r} = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}}\vec{i} + \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}}\vec{j} + \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}\vec{k}$$ This means the unit vector has the same direction as \(\vec{r}\) but a length of 1. 3. **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 their corresponding components are equal: $$\vec{r}_1 = \vec{r}_2 \implies x_1 = x_2, \quad y_1 = y_2, \quad z_1 = z_2$$ 4. **Zero Vector:** A zero vector \(\vec{0}\) has all components zero: $$\vec{0} = 0\vec{i} + 0\vec{j} + 0\vec{k}$$ It has zero magnitude and no direction. 5. **Negative of a Vector:** For a vector \(\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}\), its negative \(-\vec{r}\) is: $$-\vec{r} = (-x)\vec{i} + (-y)\vec{j} + (-z)\vec{k}$$ This vector points in the opposite direction to \(\vec{r}\). 6. **Scalar Multiplication:** Multiplying a vector \(\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}\) by a scalar \(\lambda\) scales each component: $$\lambda \vec{r} = (\lambda x)\vec{i} + (\lambda y)\vec{j} + (\lambda z)\vec{k}$$ This changes the magnitude of the vector but not its direction unless \(\lambda\) is negative, which reverses direction. 7. **Parallel Vectors:** Two non-zero 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 parallel if there exists a non-zero scalar \(\lambda\) such that: $$\vec{r}_1 = \lambda \vec{r}_2$$ Expanding: $$x_1 = \lambda x_2, \quad y_1 = \lambda y_2, \quad z_1 = \lambda z_2$$ This means the vectors have the same or exactly opposite direction. **Summary:** - Unit vectors have magnitude 1 and point in the direction of the original vector. - Equal vectors have identical components. - Zero vector has all zero components. - Negative vector reverses direction. - Scalar multiplication scales vector magnitude. - Parallel vectors differ by a scalar multiple. These definitions form the foundation for vector operations in 3D space.