Chapter 9, Problem 1PE

(a)

To determine

To find:a vector

Explanation of Solution

A Vector is a quantity that has both magnitude and direction. A vector PQ is represented as a directed line segment and an arrow with initial point $P$ and terminal point $Q$ .

  

A vector is indicated by writing an arrow over the designated points $P$ and $Q$ . The order of letters is important.

Vector PQ is the vector from $P$ to $Q$ and PQ is the vector from $Q$ to $P$

Conclusion:

Therefore, a Vector is a quantity that has both magnitude and direction.

(b)

To determine

To find: a scalar.

Explanation of Solution

A scalar is a quantity that has only magnitude, not direction. In context of vectors, a scalar is described as a real number.

For example, $\Vert PQ\Vert$ is a scalar quantity.

Conclusion:

Therefore, a scalar is a quantity that has only magnitude, not direction.

(c)

To determine

To give: an algebraic and a geometric interpretation of multiplication of a vector by a scalar.

Explanation of Solution

When a scalar number c is multiplied by a vector v , the scalar multiple cv will be parallel to the vector v and has length $\left|c\right|$ times that of v .

The scalar multiple cv points in the same direction as v if $c>0$ and in opposite direction if $c<0$ .

Geometrically, assume that vector v has a magnitude of 3 units and its direction is from point P to point Q .

  

The scalar multiple 2v will have a magnitude of $2\times 3=6units$ and its direction will be parallel to the vector v .

  

  

Conclusion:

Therefore,when a scalar number c is multiplied by a vector v , the scalar multiple cv will be parallel to the vector v and has length $\left|c\right|$ times that of v .

Geometrically, the scalar multiple 2v will have a magnitude of $2\times 3=6units$ and its direction will be parallel to the vector v .

(d)

To determine

To find: parallelogram rule for vector sums

Explanation of Solution

The Parallelogram rule states that $u+v$ is the diagonal of the parallelogram formed with sides u and v .

  

Conclusion:

Therefore, the Parallelogram rule states that $u+v$ is the diagonal of the parallelogram formed with sides u and v .