Chapter 11, Problem 1RE

(a) What is the difference between a vector and a scalar? Give a physical example of each.

(b) How can you determine whether or not two vectors are orthogonal?

(c) How can you determine whether or not two vectors are parallel?

(d) How can you determine whether or not three vectors with a common initial point lie in the same plane in 3-space?

To determine

The difference between a vector and a scalar by giving a physical example.

Answer to Problem 1RE

A scalar quantity is the quantity that has only magnitude and vector quantity is the quantity that has magnitude as well as direction.

Explanation of Solution

Scalar: A scalar quantity is the quantity that has only magnitude.

For example: time, speed, temperature, and volume.

Vector: A vector quantity is the quantity that has magnitude as well as direction.

For example: displacement, velocity, acceleration, and force.

To determine

Whether the two vectors are orthogonal or not.

Answer to Problem 1RE

The vectors, $a$ and $b$ , are orthogonal if and only if $a\cdot b=0$ .

Explanation of Solution

The two vectors are orthogonal if and only if their dot product is zero.

Therefore, the vectors, $a$ and $b$ , are orthogonal if and only if $a\cdot b=0$ .

To determine

Whether the two vectors are parallel or not.

Answer to Problem 1RE

The vectors, $a$ and $b$ , are parallel if and only if $a\times b=0$ .

Explanation of Solution

The two vectors are parallel if and only if their cross product is a zero vector.

Therefore, the vectors, $a$ and $b$ , are parallel if and only if $a\times b=0$ .

To determine

Whether or not the three vectors with a common initial point lie in the same plane in the 3-space.

Answer to Problem 1RE

The vectors, $a$ and $b$ , are coplanar if and only if $a\cdot \left(b\times c\right)=0$ .

Explanation of Solution

Given Information:

The three vectors with a common initial point lie in the same plane in the 3-space.

Three vectors with a common initial point lie in a plane if and only if their scalar triple product is zero.

Therefore, the vectors, $a$ and $b$ , are coplanar if and only if $a\cdot \left(b\times c\right)=0$ .