7.7 Applications of Dot and Cross Product
Chapter
Chapter 7
Section
7.7
Solutions 19 Videos

Calculate |\vec{a} \times \vec{b}|, where \vec{a} = (1, 2, 1) and \vec{b} = (2, 4, 2).

0.27mins
Q2a

If \vec{a} and \vec{b} represent the sides of a parallelogram, explain why your answer for part a. makes sense, in terms of the formula for the area of a parallelogram.

1.08mins
Q2b

Calculate the amount of work done in each situation.

A stove is slid 3 m across the floor against a frictional force of 150 N.

0.18mins
Q3a

Calculate the amount of work done in each situation.

A 40 kg rock falls 40 m down a slope at an angle of 50^o to the vertical.

1.35mins
Q3b

Calculate the amount of work done in each situation.

A wagon is pulled a distance of 250 m by a force of 140 N applied at an angle of 20^o to the road.

0.44mins
Q3c

Calculate the amount of work done in each situation.

A lawnmower is pushed 500 m by a force of 100 N applied at an angle of 45^o to the horizontal.

1.06mins
Q3d

Determine each of the following

\vec{i} \times \vec{j}

0.14mins
Q4a

Determine each of the following

-\vec{i} \times \vec{j}

0.12mins
Q4b

Determine each of the following

\vec{i} \times \vec{k}

0.24mins
Q4c

Determine each of the following

-\vec{i} \times \vec{k}

0.11mins
Q4d

Calculate the area of the parallelogram formed by the following pairs of vectors:

\vec{a} = (1, 1, 0) and \vec{b} = (1, 0, 1)

0.31mins
Q5a

Calculate the area of the parallelogram formed by the following pairs of vectors:

\vec{a} = (1, -2, 3) and \vec{b} = (1, 2, 4)

1.03mins
Q5b

The area of the parallelogram formed by the vectors \vec{p} = (a, 1, -1) and \vec{q} = (1, 1,2) is \sqrt{35}. Determine the value(s) of a for which this is true.

1.53mins
Q6

In \mathbb{R}^3, points A(-2, 1, 3), B(1, 0, 1) and C(2, 3, 2) form the vertices of \triangle ABC.

By constructing position vectors \vec{AB} and \vec{AC}, determine the area of the triangle.

1.58mins
Q7a

In \mathbb{R}^3, points A(-2, 1, 3), B(1, 0, 1) and C(2, 3, 2) form the vertices of \triangle ABC.

By constructing position vectors \vec{BC} and \vec{CA}, determine the area of the triangle.

2.03mins
Q7b

A 10 N force is applied at the end of a wrench that is 14 cm long. The force makes an angle of 45^o with the wrench. Determine the magnitude of the torque of this force about the other end of the wrench.

1.20mins
Q8

Parallelogram OBCA has its sides determined by \vec{OA} = \vec{a} = (4, 2, 4) and \vec{OB} =\vec{b} = (3, 1, 4). Its fourth vertex is point C. A lien is drawn from B perpendicular to side AC of the parallelogram to intersect AC at N. Determine the length of BN.

2.39mins
Q9

For the vectors \vec{p} = (1, -2, 3), \vec{q} = (2, 1, 3), and \vec{r} = (1, 1, 0), show the following to be true.

The vector (\vec{p} \times \vec{q}) \times \vec{r} can be written as a linear combination of \vec{p} and \vec{q}.

For the vectors \vec{p} = (1, -2, 3), \vec{q} = (2, 1, 3), and \vec{r} = (1, 1, 0), show the following to be true.
(\vec{p} \times \vec{q}) \times \vec{r} = (\vec{p}\cdot \vec{r})\vec{q} - (\vec{q}\cdot \vec{r})\vec{p}