6.7 Operation with Vectors in 3D
Chapter
Chapter 6
Section
6.7
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Solutions 34 Videos

a) Write the vector \vec{OA} = (-1, 2, 4) using the standard unit vectors.

b) Determine |\vec{OA}|.

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0.41mins
Q1

Write the vector \vec{OB} = 3\vec{i} + 4\vec{j} - 4\vec{k} in component form and calculate its magnitude.

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Q2

If \vec{a} = (1, 3, -3), \vec{b} =(-3, 6 ,12), and \vec{c} = (0, 8, 1), determine |\vec{a} + \displaystyle{\frac{1}{3}}\vec{b} - \vec{c}|.

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Q3

For the vectors \vec{OA} = (-3, 4, 12) and \vec{OB} = (2, 2, -1), determine the following:

the components of vector \vec{OP}, where \vec{OP} = \vec{OA} +\vec{ OB}

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Q4a

For the vectors \vec{OA} = (-3, 4, 12) and \vec{OB} = (2, 2, -1), determine the following:

|\vec{OA}|, |\vec{OB}|, and |\vec{OP}|.

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Q4b

For the vectors \vec{OA} = (-3, 4, 12) and \vec{OB} = (2, 2, -1), determine the following:

\vec{AB} and |\vec{AB}|. What does \vec{AB} represent?

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Q4c

Given \vec{x} = (1, 4, - 1), \vec{y} = (1, 3, -2), and \vec{z} = (-2, 1, 0), determine a vector equivalent to each of the following:

\displaystyle \vec{x} -2\vec{y} - \vec{z}

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Q5a

Given \vec{x} = (1, 4, - 1), \vec{y} = (1, 3, -2), and \vec{z} = (-2, 1, 0), determine a vector equivalent to each of the following:

\displaystyle -2\vec{x} - 3\vec{y} +\vec{z}

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Q5b

Given \vec{x} = (1, 4, - 1), \vec{y} = (1, 3, -2), and \vec{z} = (-2, 1, 0), determine a vector equivalent to each of the following:

\displaystyle \frac{1}{2}\vec{x} - \vec{y} + 3\vec{z}

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Q5c

Given \vec{x} = (1, 4, - 1), \vec{y} = (1, 3, -2), and \vec{z} = (-2, 1, 0), determine a vector equivalent to each of the following:

\displaystyle 3\vec{x} + 5\vec{y} + 3\vec{z}

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Q5d

Given \vec{p} = 2\vec{i} - \vec{j} + \vec{k} and \vec{q} = -\vec{i} -\vec{j} + \vec{k}, determine the following in terms of the standard unit vectors.

\displaystyle \vec{p} + \vec{q}

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Q6a

Given \vec{p} = 2\vec{i} - \vec{j} + \vec{k} and \vec{q} = -\vec{i} -\vec{j} + \vec{k}, determine the following in terms of the standard unit vectors.

\displaystyle \vec{p} - \vec{q}

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Q6b

Given \vec{p} = 2\vec{i} - \vec{j} + \vec{k} and \vec{q} = -\vec{i} -\vec{j} + \vec{k}, determine the following in terms of the standard unit vectors.

\displaystyle 2\vec{p} - 5\vec{q}

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Q6c

Given \vec{p} = 2\vec{i} - \vec{j} + \vec{k} and \vec{q} = -\vec{i} -\vec{j} + \vec{k}, determine the following in terms of the standard unit vectors.

\displaystyle -2\vec{p} +5\vec{q}

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Q6d

If \vec{m} = 2\vec{i} -\vec{j} + \vec{k} and \vec{q} = -\vec{i} -\vec{j} + \vec{k}, determine the following in terms of the standard unit vectors.

|\vec{m}-\vec{n}|

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Q7a

If \vec{m} = 2\vec{i} -\vec{j} + \vec{k} and \vec{q} = -\vec{i} -\vec{j} + \vec{k}, determine the following in terms of the standard unit vectors.

|\vec{m}+\vec{n}|

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Q7b

If \vec{m} = 2\vec{i} -\vec{j} + \vec{k} and \vec{q} = -\vec{i} -\vec{j} + \vec{k}, determine the following in terms of the standard unit vectors.

2\vec{m} + 3\vec{n}

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Q7c

If \vec{m} = 2\vec{i} -\vec{j} + \vec{k} and \vec{q} = -\vec{i} -\vec{j} + \vec{k}, determine the following in terms of the standard unit vectors.

-5\vec{m}

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Q7d

Given \vec{x} + \vec{y} = -\vec{i} + 2\vec{j} + 5\vec{k}, and \vec{n} = -2\vec{i} + \vec{j} + 2\vec{k}, determine \vec{x} and \vec{y}.

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Q8

Three vectors, \vec{OA} = (a, b, 0), \vec{OB} = (a, 0, c), and \vec{OC} = (0, b, c), are given.

In a sentence, describe what each vector represents.

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Q9a

Three vectors, \vec{OA} = (a, b, 0), \vec{OB} = (a, 0, c), and \vec{OC} = (0, b, c), are given.

Write each of the given vectors using the standard unit vectors.

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Q9b

Three vectors, \vec{OA} = (a, b, 0), \vec{OB} = (a, 0, c), and \vec{OC} = (0, b, c), are given.

Determine a formula for each of |\vec{OA}|, |\vec{OB}|, and |\vec{OC}|.

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Q9c

Three vectors, \vec{OA} = (a, b, 0), \vec{OB} = (a, 0, c), and \vec{OC} = (0, b, c), are given.

Determine \vec{AB}. What does \vec{AB} represent?

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Q9d

Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following:

|\vec{OA}|

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0.26mins
Q10a

Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following:

|\vec{OB}|

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Q10b

Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following:

\vec{AB}

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Q10c

Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following:

|\vec{AB}|

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Q10d

Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following:

\vec{BA}

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Q10e

Given the point A(-2, -6, 3) and B(3, -4, 12), determine each of the following:

|\vec{BA}|

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Q10f

The vertices of quadrilateral ABCD are given as A(0, 3, 5), B(3, -1, 17), C(7, -3, 15), and D(4, 1, 3). Prove that ABCD is a parallelogram.

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Q11

Given 2\vec{x} + \vec{y} -2\vec{z} = \vec{0}, \vec{x} = (-1, b, c), \vec{y} = (a, -2, c), and \vec{z} = (-a, 6 ,c), determine the value of the unknowns.

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Q12

A parallelepiped is determined by the vectors \vec{OA} = (-2, 2, 5), \vec{OB} = (0, 4, 1), and \vec{OC} = (0, 5, -1).

(a) Draw a sketch of the parallelepiped former by these vectors.

(b) Determine the coordinates of tall of the vertices for the parallelepiped.

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Q13

Given the points A(-2, 1, 3) and B(4, -1, 3), determine the coordinates of the point on the x-axis that is equidistant from these two points.

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Q14

Give |\vec{a}| = 3, |\vec{b}| = 5, and |\vec{a} + \vec{b}| = 7, determine |\vec{a} - \vec{b}|.

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Q15