Chapter 6, Problem 11CT

a.

To determine

To calculate:

Value of $2v+u$ using given vectors.

Answer to Problem 11CT

Value of $2v+u$ using given vectors is $-j$ .

Explanation of Solution

Given information:

  $\begin{array}{l}u=2i+3j,\\ v=-i-2j\end{array}$

Calculation:

Given vectors are,

  $\begin{array}{l}u=2i+3j,\\ v=-i-2j\end{array}$

Related formulas are,

  $\begin{array}{l}a=a_{1}i+a_{2}j,\\ b=b_{1}i+b_{2}j\end{array}$

  $k=$ scalar.

Then equation become,

  $\begin{array}{l}a+b=\left(a_1+b_1\right)i+\left(a_2+b_2\right)j\\ a-b=\left(a_1-b_1\right)i+\left(a_2-b_2\right)j\\ ka=k{a}_{1}i+k{a}_{2}j\end{array}$

Need to find out $2v+u$ ,

  $2v+u=2\left(-i-2j\right)+\left(2i+3j\right)$

Scalar multiplication,

  $=\left(-2i-4j\right)+\left(2i+3j\right)$

Vector addition,

  $\begin{array}{l}=\left(-2i+2j\right)+\left(-4j+3j\right)\\ =-j\end{array}$

Hence , Value of $2v+u$ using given vectors is $-j$ .

b.

To determine

To calculate:

Value of $u-3v$ using given vectors.

Answer to Problem 11CT

Value of $u-3v$ using given vectors is $5j+9j$ .

Explanation of Solution

Given information:

  $\begin{array}{l}u=2i+3j,\\ v=-i-2j\end{array}$

Calculation:

Given vectors are,

  $\begin{array}{l}u=2i+3j,\\ v=-i-2j\end{array}$

Related formulas are,

  $\begin{array}{l}a=a_{1}i+a_{2}j,\\ b=b_{1}i+b_{2}j\end{array}$

  $k=$ scalar.

Then equation become,

  $\begin{array}{l}a+b=\left(a_1+b_1\right)i+\left(a_2+b_2\right)j\\ a-b=\left(a_1-b_1\right)i+\left(a_2-b_2\right)j\\ ka=k{a}_{1}i+k{a}_{2}j\end{array}$

  $u-3v$ is the difference of $u$ and $3v$ ,

  $u-3v=\left(2i+3j\right)-3\left(-i-2j\right)$

Scalar multiplication,

  $=\left(2i+3j\right)-\left(-3i-6j\right)$

Vector addition,

  $\begin{array}{l}=\left(2i-\left(-3j\right)\right)-\left(-3i-6j\right)\\ =\left(2i+3j\right)+\left(3j+6j\right)\\ =5j+9j\end{array}$

Hence , Value of $u-3v$ using given vectors is $5j+9j$ .

c.

To determine

To calculate:

Value of $5u-4v$ using given vectors.

Answer to Problem 11CT

Value of $5u-4v$ using given vectors is $14i+23j$ .

Explanation of Solution

Given information:

  $\begin{array}{l}u=2i+3j,\\ v=-i-2j\end{array}$

Calculation:

Given vectors are,

  $\begin{array}{l}u=2i+3j,\\ v=-i-2j\end{array}$

Related formulas are,

  $\begin{array}{l}a=a_{1}i+a_{2}j,\\ b=b_{1}i+b_{2}j\end{array}$

  $k=$ scalar.

Then equation become,

  $\begin{array}{l}a+b=\left(a_1+b_1\right)i+\left(a_2+b_2\right)j\\ a-b=\left(a_1-b_1\right)i+\left(a_2-b_2\right)j\\ ka=k{a}_{1}i+k{a}_{2}j\end{array}$

  $5u-4v$ is the difference of $5u$ and $4v$ ,

  $5u-4v=5\left(2i+3j\right)-4\left(-i-2j\right)$

Scalar multiplication,

  $=\left(10i+15j\right)-\left(-4i-8j\right)$

Vector addition,

  $\begin{array}{l}=\left(10i+4j\right)-\left(15j+8j\right)\\ =14i+23j\end{array}$

Hence , Value of $5u-4v$ using given vectors is $14i+23j$ .