7.4 The Dot Product of Algebraic Vectors
Chapter
Chapter 7
Section
7.4
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Lectures 5 Videos

Finding the Angle between two vectors using Dot Product ex1

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1.39mins
Finding the Angle between two vectors using Dot Product ex1

Finding Dot Products from Algebraic Vectors examples

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0.46mins
Finding Dot Products from Algebraic Vectors examples

Finding the Angle between two vectors using Dot Product ex2a

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1.26mins
Finding the Angle between two vectors using Dot Product ex2a

Finding the Angle between two vectors using Dot Product ex2b

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1.14mins
Finding the Angle between two vectors using Dot Product ex2b
Solutions 37 Videos

How many vectors are perpendicular to \vec{a} =(-1, 1)? State the components of three such vectors.

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1.33mins
Q1

For each of the following pairs of vectors, calculate the dot product and, on the basis of your result, say whether the angle between the two vectors is acute, obtuse, or 90^o.

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

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0.17mins
Q2a

For each of the following pairs of vectors, calculate the dot product and, on the basis of your result, say whether the angle between the two vectors is acute, obtuse, or 90^o.

b) \vec{a} = (2, 3,-1), \vec{b} = (4, 3, -17)

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0.20mins
Q2b

For each of the following pairs of vectors, calculate the dot product and, on the basis of your result, say whether the angle between the two vectors is acute, obtuse, or 90^o.

c) \vec{a} = (1, -2,5), \vec{b} = (3, -2, -2))

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0.31mins
Q2c

Give the components of a vector that is perpendicular to each of the following planes:

a) xy-plane

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0.19mins
Q3a

Give the components of a vector that is perpendicular to each of the following planes:

b) xz-plane

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0.27mins
Q3b

Give the components of a vector that is perpendicular to each of the following planes:

c) yz-plane

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0.21mins
Q3c

a) From the set of vectors \Big\{ (1, 2, 3), (-4, -5, -6), (4, 3, 10), (5, -3, \displaystyle{\frac{-5}{6}} \Big\}, select two pairs of vectors that are perpendicular to each other.

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1.55mins
Q4a

a) From the set of vectors \Big\{ (1, 2, 3), (-4, -5, -6), (4, 3, 10), (5, -3, \displaystyle{\frac{-5}{6}} \Big\}, select two pairs of vectors that are perpendicular to each other.

b) Are any oft these vectors collinear ? Explain.

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1.25mins
Q4b

a) Explain why it would not be possible to do this in \mathbb{R}^2 if we selected the two vectors \vec{a} = (1, -2) and \vec{b} = (1, 1).

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0.52mins
Q5a

b) Explain, in general, why it is not possible to do this if we select any two vectors in \mathbb{R}^2.

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3.34mins
Q5b

Determine the angle, to the nearest degree, between each of the following pairs of vectors:

a) \vec{a} = (5, 3) and \vec{b} = (-1, -2).

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0.43mins
Q6a

Determine the angle, to the nearest degree, between each of the following pairs of vectors:

b) \vec{a} = (-1, 4) and \vec{b} = (6, -2).

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0.35mins
Q6b

Determine the angle, to the nearest degree, between each of the following pairs of vectors:

c) \vec{a} = (2, 2, 1) and \vec{b} = (2, 1, -2).

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0.41mins
Q6c

Determine the angle, to the nearest degree, between each of the following pairs of vectors:

d) \vec{a} = (2, 3, -6) and \vec{b} = (-5, 0, 12).

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0.50mins
Q6d

Determine k, given two vectors and the angel between them.

a) \vec{a} = (-1, 2, -3) and \vec{b} = (-6k, -1, k), \theta = 90^o

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0.32mins
Q7a

Determine k, given two vectors and the angel between them.

b) \vec{a} = (1, 1) and \vec{b} = (0, k), \theta 45^o

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0.54mins
Q7b

In \mathbb{R}^2, a square is determined by the vectors \vec{i} and \vec{j}.

a) Sketch the square.

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0.42mins
Q8a

In \mathbb{R}^2, a square is determined by the vectors \vec{i} and \vec{j}.

b) Determine vector components for the two diagonals.

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0.32mins
Q8b

In \mathbb{R}^2, a square is determined by the vectors \vec{i} and \vec{j}.

c) Verify that the angle between the diagonals is 90^o.

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0.34mins
Q8c

Determine the angle, to the nearest degree, between each pair of vectors.

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

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1.36mins
Q9a

Determine the angle, to the nearest degree, between each pair of vectors.

b) \vec{a} = (\sqrt{2} -1, \sqrt{2} +1, \sqrt{2}) and \vec{b} = (1 ,1, 1)

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1.41mins
Q9b

a) For the vectors \vec{a} = (2, p, 8) and \vec{b} = (q, 4, 12), determine values of p and q so that the actors are

 i) collinear

 ii) perpendicular 

b) Are the values of p and q unique? Explain.

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2.04mins
Q10

\triangle ABC has vertices at A(2, 5), B(4, 11), and C(-1, 6). Determine the angles in this triangle.

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3.51mins
Q11

A rectangular box measuring 4 by 5 by 7 as when in the diagram at the left.

a) Determine the coordinates of each of the missing vertices.

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1.07mins
Q12a

A rectangular box measuring 4 by 5 by 7 as when in the diagram at the left.

b) Determine the angle, to the nearest degree, between \vec{AE} and \vec{BF}.

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2.20mins
Q12b

a) Given the vectors \vec{p} = (-1, 3, 0) and \vec{q} = (1, -5, 2), determine the components of a vector perpendicular to each of these vectors.

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2.00mins
Q13a

b) Given the vectors \vec{m} = (1, 3, -4) and \vec{n} = (-1 ,-2, 3), determine the components of a actor perpendicular to each of these vectors.

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1.43mins
Q13b

Find the value of p if the vectors \vec{r} = (p, p, 1) and \vec{s} = (p, -2, -3) are perpendicular to each other.

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0.32mins
Q14

a) Determine the algebraic condition such that the vectors \vec{c} = (-3, p -1) and \vec{d} = (1, -4, q) are perpendicular to each other.

b) If q = -3, what is the corresponding value of p?

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0.49mins
Q15

Given the actors \vec{r} = (1, 2, -1) and \vec{s} = (-2, -4, 2), determine the components of two veto perpendicular to each of these vectors. Explain you answer.

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1.29mins
Q16

The vectors \vec{x} = (-4, p, -2) and \vec{y} = (-2, 3, 6) are such that \cos^{-1}(\displaystyle{\frac{4}{21}}) = \theta, where \theta is the angle between \vec{x} and \vec{y}. Determine the value(s) of p.

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5.15mins
Q17

The diagonals of a parallelogram are determined by the vectors \vec{a} = 3, ,3 0) and \vec{b} = (-1, 1, -2).

a) Show that this parallelogram is a thumbs.

b) Determine vectors representing its sides and then determine the length of these sides.

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3.06mins
Q18ab

The diagonals of a parallelogram are determined by the vectors \vec{a} = 3, ,3 0) and \vec{b} = (-1, 1, -2).

c) Determine the angles in this rhombus.

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2.43mins
Q18c

The rectangle ABCD has vertices at A(-1, 2, 3) , B(2, 6, -9), and D(3, q, 8).

a) Determine the coordinates of the vertex C.

b) Determine the angle between the two diagonals of this rectangle.

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2.59mins
Q19a

A cube measures 1 by 1 by 1. A line is drawn from one vertex to a diagonally opposite vertex through the centre of the cube. This is called a body diagonal for the cube. Determine the angles between the body diagonals of the cube.

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1.55mins
Q20