Problem 12 - Contemporary Precalculus

Question

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=\langle 4,0\rangle, \mathbf{v}=\langle 1,-3\rangle$$

Step-by-Step Solution

Answer

Question: Find the following operations involving two vectors \(\mathbf{u} = \langle 4, 0 \rangle\) and \(\mathbf{v} = \langle 1, -3 \rangle\): (a) \(\mathbf{u} + \mathbf{v}\), (b) \(\mathbf{v} - \mathbf{u}\), (c) \(2\mathbf{u} - 3\mathbf{v}\). Answer: (a) \(\mathbf{u} + \mathbf{v} = \langle 5, -3 \rangle\), (b) \(\mathbf{v} - \mathbf{u} = \langle -3, -3 \rangle\), (c) \(2\mathbf{u} - 3\mathbf{v} = \langle 5, 9 \rangle\).

Step 1: Finding the sum of the vectors

To find the sum of the vectors, we add their corresponding components. So, \((u_x + v_x, u_y + v_y)\) $$\mathbf{u}+\mathbf{v}=\langle 4,0 \rangle+\langle 1, -3 \rangle=\langle 4+1,0+(-3) \rangle=\langle 5 ,-3 \rangle$$

Step 2: Finding the difference of the vectors

To find the difference of the vectors, we subtract their corresponding components. So, \((v_x - u_x, v_y - u_y)\) $$\mathbf{v}-\mathbf{u}=\langle 1, -3 \rangle-\langle 4, 0 \rangle=\langle 1-4, -3-0 \rangle=\langle -3, -3 \rangle$$

Step 3: Finding the linear combination of the vectors

First, calculate \(2\mathbf{u}\) and \(-3\mathbf{v}\) $$2\mathbf{u} = 2\langle 4, 0 \rangle = \langle 8, 0 \rangle$$ $$-3\mathbf{v} = -3\langle 1, -3 \rangle = \langle -3, 9 \rangle$$ Now, add the two resulting vectors to find the linear combination: $$2\mathbf{u}-3\mathbf{v}=\langle 8, 0 \rangle+\langle -3, 9 \rangle=\langle 8-3, 0+9 \rangle=\langle 5, 9 \rangle$$

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. To perform vector addition, simply add the corresponding components of the vectors involved. Suppose we have two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), represented in component form as \(\mathbf{u} = \langle u_x, u_y \rangle\) and \(\mathbf{v} = \langle v_x, v_y \rangle\). Then, the sum \(\mathbf{u} + \mathbf{v}\) is calculated as follows:
\[ \mathbf{u} + \mathbf{v} = \langle u_x + v_x, u_y + v_y \rangle \]

For example, if \(\mathbf{u} = \langle 4, 0 \rangle\) and \(\mathbf{v} = \langle 1, -3 \rangle\), their sum is:
\[ \mathbf{u} + \mathbf{v} = \langle 4+1, 0+(-3) \rangle = \langle 5, -3 \rangle \]
Remember:

• Each vector has direction and magnitude, meaning that vector addition takes into account these aspects.
• The order in which vectors are added doesn't matter due to the commutative property, \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\).

Vector Subtraction

Vector subtraction involves determining the vector difference by subtracting one vector from another. Similar to addition, the subtraction of vectors also occurs component-wise. Given two vectors \(\mathbf{u}\) and \(\mathbf{v}\), their difference \(\mathbf{u} - \mathbf{v}\) is calculated by:
\[ \mathbf{u} - \mathbf{v} = \langle u_x - v_x, u_y - v_y \rangle \]

For the vectors \(\mathbf{u} = \langle 4, 0 \rangle\) and \(\mathbf{v} = \langle 1, -3 \rangle\), subtract \(\mathbf{v}\) from \(\mathbf{u}\) as follows:
\[ \mathbf{v} - \mathbf{u} = \langle 1 - 4, -3 - 0 \rangle = \langle -3, -3 \rangle \]
This result, \(\langle -3, -3 \rangle\), signifies moving in the opposite direction relative to the subtraction order.

• Subtracting a vector is similar to adding its inverse.
• Vector subtraction is not commutative, meaning \(\mathbf{u} - \mathbf{v} eq \mathbf{v} - \mathbf{u}\).

Linear Combination of Vectors

A linear combination of vectors is an expression of the form \(a\mathbf{u} + b\mathbf{v}\), where \(a\) and \(b\) are scalars that stretch or shrink the vectors. This involves both scaling the vectors and then adding the results.

For instance, consider \(2\mathbf{u} - 3\mathbf{v}\) using \(\mathbf{u} = \langle 4, 0 \rangle\) and \(\mathbf{v} = \langle 1, -3 \rangle\). Calculate as follows:
- Scale \(\mathbf{u}\) by 2:
\[ 2\mathbf{u} = \langle 8, 0 \rangle \]
- Scale \(\mathbf{v}\) by -3:
\[ -3\mathbf{v} = \langle -3, 9 \rangle \]
- Add the results:
\[ 2\mathbf{u} - 3\mathbf{v} = \langle 8+(-3), 0+9 \rangle = \langle 5, 9 \rangle \]

• Linear combinations allow you to create new vectors that lie in the same plane or space defined by the original vectors.
• It’s crucial in vector spaces used in various fields such as physics and computer graphics.