Chapter 9, Problem 1RCC

(a)

To determine

To find: The difference between a scalar and a vector.

Answer to Problem 1RCC

The difference between a scalar and a vector is that a scalar is a quantity that has magnitude only and there is no need of direction and a vector is a quantity that has both magnitude and direction.

Explanation of Solution

Given information:

Calculation:

Scalar:- A scalar is a quantity that has magnitude only and there is no need of direction.

For example: Length Mass Area: Temperature etc. these are the quantities that are determined completely by their magnitude only.

Vector:- A vectoris a quantity that has both magnitude and direction. The length of a vector representsthe magnitude and the arrow represents direction.

For example: Velocity. Acceleration, Force etc. these are the quantities that are determined by their magnitude and the direction

(b)

To determine

To find: The method to add two vectors.

Explanation of Solution

Calculation:

To add the two vectors $uandv$ , we sketch the vectors equal to u and v with the initial point of one at the terminal point of the other as shown below. If the displacement $u\text{ =}\text{AB}$ is followed by $\text{v =}\text{BC}$ then the sum of the vectors u and v is represented by the displacement $\text{AC =}AB+\text{BC}$ .

(c)

To determine

To find: The method to subtract two vectors.

Explanation of Solution

Calculation:

To subtract the two vectors $uandv$ , we sketch the -vthat has same length as v but with theopposite direction. And then we add the two vectors u and −v as $u+(-v)=u-v$ Thus, the vector $u-v$ denotes the subtraction of the vectors u and that is represented by the other diagonal of the parallelogram formed by u and v as shown below.

(d)

To determine

To find: The method to multiply two vectors.

Explanation of Solution

Calculation:

We need to draw a diagram to show how to multiply a vector by the scalar $2,\frac{1}{2},-2and-\frac{1}{2}$ .

If a real number and $v$ is a vector, then we define a new vector $av$ as follows. The vector $av$ has the magnitude $\left|a\right|\left|v\right|$ and has the same direction as $v$ if $a>0$ and the opposite direction if $a<0$ This process is called multiplication of a vector by a scalar.

To multiply a vector $v$ by the scalar 2.we stretch double the length of the vector $v$ in the same direction as the vector v as shown below.

To multiply a vector $v$ by the scalar $\frac{1}{2}$ , we shrink half of the length of the vector $v$ in the same direction as the vector v as shown below.

To multiply a vector $v$ by the scalar -2.we stretch double the length of the vector $v$ in the same direction as the vector v as shown below.

To multiply a vector $v$ by the scalar $\frac{1}{2}$ , we shrink half of the length of the vector $v$ in the same direction as the vector v as shown below.