1. **Problem Statement:** We want to understand the components of a vector and how to find the magnitude of a vector in 3D space. 2. **Vector Components:** A vector \( \mathbf{r} \) in 3D space can be written as \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), where: - \( x \) is the component along the x-axis, - \( y \) is the component along the y-axis, - \( z \) is the component along the z-axis. These components tell us how far the vector extends in each direction. 3. **Analytical Representation:** Writing the vector as \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) is called the analytical representation of the vector. 4. **Magnitude of the Vector:** The magnitude (or length) of the vector \( \mathbf{r} \) is found using the Pythagorean theorem in 3D: $$|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$$ This comes from: - First finding the length of the projection on the xy-plane: \( |OQ| = \sqrt{x^2 + y^2} \) - Then combining it with the z-component using: $$|\mathbf{r}|^2 = |OQ|^2 + z^2 = (\sqrt{x^2 + y^2})^2 + z^2 = x^2 + y^2 + z^2$$ 5. **Summary:** The components \( x, y, z \) describe how much the vector points in each axis direction, and the magnitude tells us the total length of the vector in space. This helps us understand vectors in 3D by breaking them down into simpler parts along each axis and then combining those parts to find the overall size. **Final answer:** The magnitude of the vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) is $$|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$$