The given expressions

Question

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Answer 1

Question

Chapter 12, Problem 1CRE

To determine

The given expressions

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

Explanation of Solution

Given:

$v = 〈 - 2, 5 〉$
$w = 〈 3, - 2 〉$

Key concepts applied:

Vector operations

Vector addition

To add the vectors $〈 x_{1}, y_{1} 〉$ and $〈 x_{2}, y_{2} 〉$ , we add the corresponding components from each vector: $〈 x_{1} + x_{2}, y_{1} + y_{2} 〉$ .

Scalar multiplication

When we multiply a vector by a scalar, the result is a vector. When a vector is multiplied by a scalar i.e. a regular number, we have to multiply it on every component in the vector to get the resultant vector.

The objective is to calculate $5 w - 3 v$ and $5 v - 3 w$ .

Calculations:

Description	$v = 〈 - 2, 5 〉$ $w = 〈 3, - 2 〉$ To determine $5 w - 3 v$ and $5 v - 3 w$ .
Step 1. Substitute the vectors in the given expression	$\begin{array}{l} 5 w - 3 v \\ \Rightarrow 5 〈 3, - 2 〉 - 3 〈 - 2, 5 〉 \end{array}$	$\begin{array}{l} 5 v - 3 w \\ \Rightarrow 5 〈 - 2, 5 〉 - 3 〈 3, - 2 〉 \end{array}$
Step 2. Do scalar multiplication by multiplying it on every component in the vector	$\Rightarrow 〈 15, - 10 〉 + 〈 6, - 15 〉$	$\Rightarrow 〈 - 10, 25 〉 + 〈 - 9, 6 〉$
Step 3. Do vector addition by adding the corresponding components from each vector	$\begin{array}{l} \Rightarrow 〈 15 + 6, - 10 - 15 〉 \\ \Rightarrow 〈 21, - 25 〉 \end{array}$	$\begin{array}{l} \Rightarrow 〈 - 10 - 9, 25 + 6 〉 \\ \Rightarrow 〈 - 19, 31 〉 \end{array}$

Conclusion:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

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Answer 2

Question

Chapter 12, Problem 1CRE

To determine

The given expressions

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

Explanation of Solution

Given:

$v = 〈 - 2, 5 〉$
$w = 〈 3, - 2 〉$

Key concepts applied:

Vector operations

Vector addition

To add the vectors $〈 x_{1}, y_{1} 〉$ and $〈 x_{2}, y_{2} 〉$ , we add the corresponding components from each vector: $〈 x_{1} + x_{2}, y_{1} + y_{2} 〉$ .

Scalar multiplication

When we multiply a vector by a scalar, the result is a vector. When a vector is multiplied by a scalar i.e. a regular number, we have to multiply it on every component in the vector to get the resultant vector.

The objective is to calculate $5 w - 3 v$ and $5 v - 3 w$ .

Calculations:

Description	$v = 〈 - 2, 5 〉$ $w = 〈 3, - 2 〉$ To determine $5 w - 3 v$ and $5 v - 3 w$ .
Step 1. Substitute the vectors in the given expression	$\begin{array}{l} 5 w - 3 v \\ \Rightarrow 5 〈 3, - 2 〉 - 3 〈 - 2, 5 〉 \end{array}$	$\begin{array}{l} 5 v - 3 w \\ \Rightarrow 5 〈 - 2, 5 〉 - 3 〈 3, - 2 〉 \end{array}$
Step 2. Do scalar multiplication by multiplying it on every component in the vector	$\Rightarrow 〈 15, - 10 〉 + 〈 6, - 15 〉$	$\Rightarrow 〈 - 10, 25 〉 + 〈 - 9, 6 〉$
Step 3. Do vector addition by adding the corresponding components from each vector	$\begin{array}{l} \Rightarrow 〈 15 + 6, - 10 - 15 〉 \\ \Rightarrow 〈 21, - 25 〉 \end{array}$	$\begin{array}{l} \Rightarrow 〈 - 10 - 9, 25 + 6 〉 \\ \Rightarrow 〈 - 19, 31 〉 \end{array}$

Conclusion:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

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Answer 3

Question

Chapter 12, Problem 1CRE

To determine

The given expressions

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

Explanation of Solution

Given:

$v = 〈 - 2, 5 〉$
$w = 〈 3, - 2 〉$

Key concepts applied:

Vector operations

Vector addition

To add the vectors $〈 x_{1}, y_{1} 〉$ and $〈 x_{2}, y_{2} 〉$ , we add the corresponding components from each vector: $〈 x_{1} + x_{2}, y_{1} + y_{2} 〉$ .

Scalar multiplication

When we multiply a vector by a scalar, the result is a vector. When a vector is multiplied by a scalar i.e. a regular number, we have to multiply it on every component in the vector to get the resultant vector.

The objective is to calculate $5 w - 3 v$ and $5 v - 3 w$ .

Calculations:

Description	$v = 〈 - 2, 5 〉$ $w = 〈 3, - 2 〉$ To determine $5 w - 3 v$ and $5 v - 3 w$ .
Step 1. Substitute the vectors in the given expression	$\begin{array}{l} 5 w - 3 v \\ \Rightarrow 5 〈 3, - 2 〉 - 3 〈 - 2, 5 〉 \end{array}$	$\begin{array}{l} 5 v - 3 w \\ \Rightarrow 5 〈 - 2, 5 〉 - 3 〈 3, - 2 〉 \end{array}$
Step 2. Do scalar multiplication by multiplying it on every component in the vector	$\Rightarrow 〈 15, - 10 〉 + 〈 6, - 15 〉$	$\Rightarrow 〈 - 10, 25 〉 + 〈 - 9, 6 〉$
Step 3. Do vector addition by adding the corresponding components from each vector	$\begin{array}{l} \Rightarrow 〈 15 + 6, - 10 - 15 〉 \\ \Rightarrow 〈 21, - 25 〉 \end{array}$	$\begin{array}{l} \Rightarrow 〈 - 10 - 9, 25 + 6 〉 \\ \Rightarrow 〈 - 19, 31 〉 \end{array}$

Conclusion:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

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Answer 4

Question

Chapter 12, Problem 1CRE

To determine

The given expressions

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

Explanation of Solution

Given:

$v = 〈 - 2, 5 〉$
$w = 〈 3, - 2 〉$

Key concepts applied:

Vector operations

Vector addition

To add the vectors $〈 x_{1}, y_{1} 〉$ and $〈 x_{2}, y_{2} 〉$ , we add the corresponding components from each vector: $〈 x_{1} + x_{2}, y_{1} + y_{2} 〉$ .

Scalar multiplication

When we multiply a vector by a scalar, the result is a vector. When a vector is multiplied by a scalar i.e. a regular number, we have to multiply it on every component in the vector to get the resultant vector.

The objective is to calculate $5 w - 3 v$ and $5 v - 3 w$ .

Calculations:

Description	$v = 〈 - 2, 5 〉$ $w = 〈 3, - 2 〉$ To determine $5 w - 3 v$ and $5 v - 3 w$ .
Step 1. Substitute the vectors in the given expression	$\begin{array}{l} 5 w - 3 v \\ \Rightarrow 5 〈 3, - 2 〉 - 3 〈 - 2, 5 〉 \end{array}$	$\begin{array}{l} 5 v - 3 w \\ \Rightarrow 5 〈 - 2, 5 〉 - 3 〈 3, - 2 〉 \end{array}$
Step 2. Do scalar multiplication by multiplying it on every component in the vector	$\Rightarrow 〈 15, - 10 〉 + 〈 6, - 15 〉$	$\Rightarrow 〈 - 10, 25 〉 + 〈 - 9, 6 〉$
Step 3. Do vector addition by adding the corresponding components from each vector	$\begin{array}{l} \Rightarrow 〈 15 + 6, - 10 - 15 〉 \\ \Rightarrow 〈 21, - 25 〉 \end{array}$	$\begin{array}{l} \Rightarrow 〈 - 10 - 9, 25 + 6 〉 \\ \Rightarrow 〈 - 19, 31 〉 \end{array}$

Conclusion:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

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Answer 5

Chapter 12, Problem 1CRE

To determine

The given expressions

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

Explanation of Solution

Given:

$v = 〈 - 2, 5 〉$
$w = 〈 3, - 2 〉$

Key concepts applied:

Vector operations

Vector addition

To add the vectors $〈 x_{1}, y_{1} 〉$ and $〈 x_{2}, y_{2} 〉$ , we add the corresponding components from each vector: $〈 x_{1} + x_{2}, y_{1} + y_{2} 〉$ .

Scalar multiplication

When we multiply a vector by a scalar, the result is a vector. When a vector is multiplied by a scalar i.e. a regular number, we have to multiply it on every component in the vector to get the resultant vector.

The objective is to calculate $5 w - 3 v$ and $5 v - 3 w$ .

Calculations:

Description	$v = 〈 - 2, 5 〉$ $w = 〈 3, - 2 〉$ To determine $5 w - 3 v$ and $5 v - 3 w$ .
Step 1. Substitute the vectors in the given expression	$\begin{array}{l} 5 w - 3 v \\ \Rightarrow 5 〈 3, - 2 〉 - 3 〈 - 2, 5 〉 \end{array}$	$\begin{array}{l} 5 v - 3 w \\ \Rightarrow 5 〈 - 2, 5 〉 - 3 〈 3, - 2 〉 \end{array}$
Step 2. Do scalar multiplication by multiplying it on every component in the vector	$\Rightarrow 〈 15, - 10 〉 + 〈 6, - 15 〉$	$\Rightarrow 〈 - 10, 25 〉 + 〈 - 9, 6 〉$
Step 3. Do vector addition by adding the corresponding components from each vector	$\begin{array}{l} \Rightarrow 〈 15 + 6, - 10 - 15 〉 \\ \Rightarrow 〈 21, - 25 〉 \end{array}$	$\begin{array}{l} \Rightarrow 〈 - 10 - 9, 25 + 6 〉 \\ \Rightarrow 〈 - 19, 31 〉 \end{array}$

Conclusion:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

Answer 6

Chapter 12, Problem 1CRE

To determine

The given expressions

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

Explanation of Solution

Given:

$v = 〈 - 2, 5 〉$
$w = 〈 3, - 2 〉$

Key concepts applied:

Vector operations

Vector addition

To add the vectors $〈 x_{1}, y_{1} 〉$ and $〈 x_{2}, y_{2} 〉$ , we add the corresponding components from each vector: $〈 x_{1} + x_{2}, y_{1} + y_{2} 〉$ .

Scalar multiplication

When we multiply a vector by a scalar, the result is a vector. When a vector is multiplied by a scalar i.e. a regular number, we have to multiply it on every component in the vector to get the resultant vector.

The objective is to calculate $5 w - 3 v$ and $5 v - 3 w$ .

Calculations:

Description	$v = 〈 - 2, 5 〉$ $w = 〈 3, - 2 〉$ To determine $5 w - 3 v$ and $5 v - 3 w$ .
Step 1. Substitute the vectors in the given expression	$\begin{array}{l} 5 w - 3 v \\ \Rightarrow 5 〈 3, - 2 〉 - 3 〈 - 2, 5 〉 \end{array}$	$\begin{array}{l} 5 v - 3 w \\ \Rightarrow 5 〈 - 2, 5 〉 - 3 〈 3, - 2 〉 \end{array}$
Step 2. Do scalar multiplication by multiplying it on every component in the vector	$\Rightarrow 〈 15, - 10 〉 + 〈 6, - 15 〉$	$\Rightarrow 〈 - 10, 25 〉 + 〈 - 9, 6 〉$
Step 3. Do vector addition by adding the corresponding components from each vector	$\begin{array}{l} \Rightarrow 〈 15 + 6, - 10 - 15 〉 \\ \Rightarrow 〈 21, - 25 〉 \end{array}$	$\begin{array}{l} \Rightarrow 〈 - 10 - 9, 25 + 6 〉 \\ \Rightarrow 〈 - 19, 31 〉 \end{array}$

Conclusion:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

Answer 7

Chapter 12, Problem 1CRE

To determine

The given expressions

Answer 8

Expert Solution & Answer

Answer to Problem 1CRE

Solution:

The given expressions were calculated and the results are as below

$5 w - 3 v = 〈 21, - 25 〉$

$5 v - 3 w = 〈 - 19, 31 〉$

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

Given:

$v = 〈 - 2, 5 〉$
$w = 〈 3, - 2 〉$

Key concepts applied:

Vector operations

Vector addition

To add the vectors $〈 x_{1}, y_{1} 〉$ and $〈 x_{2}, y_{2} 〉$ , we add the corresponding components from each vector: $〈 x_{1} + x_{2}, y_{1} + y_{2} 〉$ .

Scalar multiplication

When we multiply a vector by a scalar, the result is a vector. When a vector is multiplied by a scalar i.e. a regular number, we have to multiply it on every component in the vector to get the resultant vector.

The objective is to calculate $5 w - 3 v$ and $5 v - 3 w$ .

Calculations:

Description	$v = 〈 - 2, 5 〉$ $w = 〈 3, - 2 〉$ To determine $5 w - 3 v$ and $5 v - 3 w$ .
Step 1. Substitute the vectors in the given expression	$\begin{array}{l} 5 w - 3 v \\ \Rightarrow 5 〈 3, - 2 〉 - 3 〈 - 2, 5 〉 \end{array}$	$\begin{array}{l} 5 v - 3 w \\ \Rightarrow 5 〈 - 2, 5 〉 - 3 〈 3, - 2 〉 \end{array}$
Step 2. Do scalar multiplication by multiplying it on every component in the vector	$\Rightarrow 〈 15, - 10 〉 + 〈 6, - 15 〉$	$\Rightarrow 〈 - 10, 25 〉 + 〈 - 9, 6 〉$
Step 3. Do vector addition by adding the corresponding components from each vector	$\begin{array}{l} \Rightarrow 〈 15 + 6, - 10 - 15 〉 \\ \Rightarrow 〈 21, - 25 〉 \end{array}$	$\begin{array}{l} \Rightarrow 〈 - 10 - 9, 25 + 6 〉 \\ \Rightarrow 〈 - 19, 31 〉 \end{array}$