Solved Examples: Vector Addition and Scalar Multiplication in R^3 and R^4

📚 Quick Study Guide

• ➕ Vector Addition in $\mathbb{R}^3$ and $\mathbb{R}^4$: To add vectors, simply add their corresponding components. For example, if $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$, then $\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3)$. This extends similarly to $\mathbb{R}^4$.
• 🔢 Scalar Multiplication in $\mathbb{R}^3$ and $\mathbb{R}^4$: To multiply a vector by a scalar, multiply each component of the vector by the scalar. If $c$ is a scalar and $\mathbf{u} = (u_1, u_2, u_3)$, then $c\mathbf{u} = (cu_1, cu_2, cu_3)$. This also applies to vectors in $\mathbb{R}^4$.
• ⚖️ Properties of Vector Addition and Scalar Multiplication: These operations satisfy properties like commutativity, associativity, distributivity, and the existence of an additive identity (the zero vector). These properties hold true in both $\mathbb{R}^3$ and $\mathbb{R}^4$.
• 🧭 Vectors in $\mathbb{R}^3$: These are ordered triples, commonly represented as $(x, y, z)$, corresponding to points in 3D space.
• ⬆️ Vectors in $\mathbb{R}^4$: These are ordered quadruples, represented as $(w, x, y, z)$. While harder to visualize, the algebraic operations are analogous to those in lower dimensions.

Practice Quiz

1. Question 1: What is the result of adding the vectors $\mathbf{u} = (1, 2, 3)$ and $\mathbf{v} = (4, -5, 6)$ in $\mathbb{R}^3$?
1. A) $(5, -3, 9)$
2. B) $(3, 7, -3)$
3. C) $(-3, -7, -9)$
4. D) $(5, 7, 9)$
2. Question 2: If $\mathbf{w} = (2, 0, -1, 3)$ and $c = -2$, what is $c\mathbf{w}$?
1. A) $(-4, 0, 2, -6)$
2. B) $(4, 0, -2, 6)$
3. C) $(-4, 0, -2, -6)$
4. D) $(0, 0, 0, 0)$
3. Question 3: Which property is demonstrated by the equation $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$?
1. A) Associativity
2. B) Distributivity
3. C) Commutativity
4. D) Identity
4. Question 4: Given $\mathbf{a} = (1, 1, 1, 1)$ and $\mathbf{b} = (2, 2, 2, 2)$, find $\mathbf{a} + \mathbf{b}$.
1. A) $(3, 3, 3, 3)$
2. B) $(1, 1, 1, 1)$
3. C) $(2, 2, 2, 2)$
4. D) $(0, 0, 0, 0)$
5. Question 5: If $5\mathbf{u} = (5, 10, -15)$ what is $\mathbf{u}$?
1. A) $(1, 2, -3)$
2. B) $(25, 50, -75)$
3. C) $(0, 0, 0)$
4. D) $(5, 10, -15)$
6. Question 6: What is the additive identity vector in $\mathbb{R}^4$?
1. A) $(1, 1, 1, 1)$
2. B) $(0, 0, 0, 0)$
3. C) $(1, 0, 0, 0)$
4. D) $(0, 1, 0, 0)$
7. Question 7: Let $\mathbf{x} = (3, -2, 1)$ and $\mathbf{y} = (-1, 0, 4)$. Find $2\mathbf{x} - \mathbf{y}$.
1. A) $(7, -4, -2)$
2. B) $(5, -4, -2)$
3. C) $(7, -4, 2)$
4. D) $(5, -2, 5)$