Chapter 9.1, Problem 1E

1. (a) A vector in the plane is a line segment with an assigned direction. In Figure I below, the vector u has initial point _____ and terminal point _____. Sketch the vectors 2u and u + v.
2. (b) A vector in a coordinate plane is expressed by using components. In Figure II below, the vector u has initial point $(\boxed{},\boxed{})$ and terminal point $(\boxed{},\boxed{})$. In component form we write $u=\langle \boxed{},\boxed{}\rangle$, and $v=\langle \boxed{},\boxed{}\rangle$. Then $2u=\langle \boxed{},\boxed{}\rangle$ and $u+v=\langle \boxed{},\boxed{}\rangle$.

To determine

To fill: A vector in the plane is a line segment with an assigned direction. In Figure I, The vector has initial point _____ and terminal point ________. Sketch the vector 2$u$ and $u+v$.

Answer to Problem 1E

A vector in the plane is a line segment with an assigned direction. In Figure I, The vector has initial point A and terminal point B.

Explanation of Solution

From the given figure I, it is observed that the vector $u$ with an arrow to show the direction. The arrow starts at the initial point and ends at the terminal point. The given vector is denoted by $u=\overrightarrow{AB}$.

Hence, the initial point is A and terminal point is B for the vector $u$.

The vector $2u$ is double the magnitude of vector $u$ in the same direction and the vector $u+v$ is the diagonal of the parallelogram formed by the vectors $u$ and $v$ starting at the initial point.

Figure 1 shows the vectors $2u$ and $u+v$. The direction of $2u$ is same as the direction of u.

To determine

To fill: A vector in a coordinate plane is expressed by using components. In Figure II, the vector has initial point ____ and the terminal point _____. In component form we write u = ____ and v = _____. Then 2u = ___ and $u+v$ = ___.

Answer to Problem 1E

A vector in a coordinate plane is expressed by using components. In Figure II, the vector has initial point $\left(2,1\right)$ and the terminal point $\left(4,3\right)$. In component form we write u = $\langle 2,2\rangle$ and v = $\langle -3,6\rangle$. Then 2u = $\langle 4,4\rangle$ and $u+v$ = $\langle -1,8\rangle$.

Explanation of Solution

Formula used:

The formula to calculate the component form of the vector $u$ in the plane with initial point $P\left(x_1,y_1\right)$ and $Q\left(x_2,y_2\right)$ is ,

$u=\langle x_2-x_1,y_2-y_1\rangle$ (1)

Calculation:

From the given Figure II, it is observed that the initial and terminal points of the vector $u$ are $\left(2,1\right)$ and $\left(4,3\right)$ and initial and terminal points of the vector $v$ are $\left(6,1\right)$ and $\left(3,7\right)$.

The initial point is $\left(2,1\right)$ and terminal point is $\left(4,3\right)$ for the vector $u$.

Substitute 2 for $x_1$, 4 for $x_2$, 1 for $y_1$ and 3 for $y_2$ in equation (1) to calculate $u$.

$\begin{array}{c}u=\langle 4-2,3-1\rangle \\ =\langle 2,2\rangle \end{array}$

Substitute 6 for $x_1$, 3 for $x_2$, 1 for $y_1$, 7 for $y_2$ and $v$ for $u$ in the equation (1) to obtain $v$.

$\begin{array}{c}v=\langle 3-6,7-1\rangle \\ =\langle -3,6\rangle \end{array}$

In component form the value of $u$ is $\langle 2,2\rangle$ and the value of $v$ is $\langle -3,6\rangle$.

Substitute $\langle 2,2\rangle$ for $u$ and $\langle -3,6\rangle$ for $v$ to calculate $2u$ and $u+v$.

$\begin{array}{c}2u=2\langle 2,2\rangle \\ =\langle 4,4\rangle \end{array}$

Thus,

$\begin{array}{c}u+v=\langle 2,2\rangle +\langle -3,6\rangle \\ =\langle 2-3,2+6\rangle \\ =\langle -1,8\rangle \end{array}$

Hence, the initial point is $\left(2,1\right)$ and terminal point is $\left(4,3\right)$ for the vector $u$, the value of $u$ is $\langle 2,2\rangle$ and the value of $v$ is $\langle -3,6\rangle$ and $2u=\langle 4,4\rangle$ and $u+v=\langle -1,8\rangle$.