6.4 Properties of Vectors
Chapter
Chapter 6
Section
6.4
Lectures 2 Videos
Solutions 17 Videos

If * is an operation on a set, S, the element x, such that a*x =a, is called the identity element for the operation *.

a) For the addition of numbers, what is the identity element?

b) For the multiplication of numbers, what is the identity element?

c) For the addition of vectors, what is the identity element?

d) For scalar multiplication, what is the identity element?

1.03mins
Q1

Illustrate the commutative law for two vectors that are perpendicular.

0.53mins
Q2

Redraw the following three vectors and illustrate the associate law. 1.27mins
Q3

With the use of a diagram, show that the distributive law, k(\vec{a} + \vec{b}) =k\vec{a} + k\vec{b} , holds where k <0, k \in \mathbb{R}.

2.37mins
Q4

Using the given diagram, show that the following is true.

 \vec{PQ} =

=( \vec{RQ} + \vec{SR}) + \vec{TS} + \vec{PT}

=\vec{RQ} +( \vec{SR}) + \vec{TS} ) + \vec{PT}

=\vec{RQ} + \vec{SR}) + (\vec{TS} + \vec{PT}) 1.31mins
Q5

ABCDEFGH is a rectangular prism.

Write a single vector that is equivalent to \vec{EG} + \vec{GH} + \vec{HD} + \vec{DC}

0.39mins
Q6a

ABCDEFGH is a rectangular prism.

Write a vector that is equivalent to \vec{EG} + \vec{GD} + \vec{DE}

0.28mins
Q6b

ABCDEFGH is a rectangular prism.

Is it true that |\vec{HB}| = |\vec{GA}|? Explain.

3.33mins
Q6c

Write the following vector in simplified form:

3(\vec{a} -2\vec{b} -5\vec{c}) - 3(2\vec{a} -4\vec{b} + 2\vec{c}) - (\vec{a}-3\vec{b} +3\vec{c})

1.14mins
Q7

If \vec{a} = 3\vec{i} -4\vec{j} + \vec{k} and \vec{b} = -2\vec{i} + 3\vec{j}-\vec{k}, express each of the following in terms of \vec{i}, \vec{j}, and \vec{k}.

2\vec{a} -3\vec{b}

0.42mins
Q8a

If \vec{a} = 3\vec{i} -4\vec{j} + \vec{k} and \vec{b} = -2\vec{i} + 3\vec{j}-\vec{k}, express each of the following in terms of \vec{i}, \vec{j}, and \vec{k}.

\vec{a} + 5\vec{b}

0.36mins
Q8b

If \vec{a} = 3\vec{i} -4\vec{j} + \vec{k} and \vec{b} = -2\vec{i} + 3\vec{j}-\vec{k}, express each of the following in terms of \vec{i}, \vec{j}, and \vec{k}.

2(\vec{a} -\vec{b}) -3(-2\vec{a} - 7\vec{b})

1.20mins
Q8c

If 2\vec{x} + 3\vec{y} = \vec{a} and -\vec{x} + 5\vec{y} = 6\vec{b}, express \vec{x} and \vec{y} in terms of \vec{a} and \vec{b}.

3.03mins
Q9

If \vec{x} = \displaystyle{\frac{2}{3}}\vec{y} + \displaystyle{\frac{1}{3}}\vec{z}, \vec{x} - \vec{y } = \vec{a}, and \vec{y} - \vec{z} = \vec{b}, show that \vec{a} = -\displaystyle{\frac{1}{3}}\vec{b}.

1.59mins
Q10

A cube is constructed form the three vectors \vec{a}, \vec{b}, and \vec{c}, as shown below. Express each of the diagonals \vec{AG}, \vec{BH}, \vec{CE}, and \vec{DF} in terms of \vec{a}, \vec{b}, and \vec{c}.

1.02mins
Q11a

A cube is constructed form the three vectors \vec{a}, \vec{b}, and \vec{c}, as shown below. Is |\vec{AG}|= |\vec{BH}|? Explain.

In the trapezoid TXYZ, \vec{TX} = 2\vec{ZY}. If the diagonals meet at O, find an expression for \vec{TO} in terms of \vec{TX} and \vec{TZ}.