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Exam 9: Vectors in Two and Three Dimensions

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Question 1

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Determine whether $\mathbf { u } = \left\langle - \frac { 1 } { 2 } , 2 \right\rangle$ is orthogonal to $\mathbf { v } = \left\langle - 2 , \frac { 1 } { 2 } \right\rangle$ .

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\(\mathbf { u } \cdot \mathbf { v } = \left( - \frac { 1 } { 2 } \right) ( - 2 ) + ( 2 ) \left( \frac { 1 } { 2 } \right) = 2\) , not orthogonal

Question 2

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Find a vector that is perpendicular to the plane passing through the three given points. $P ( 3,0,1 ) , Q ( 0,2 , - 5 ) , R ( - 2,0,5 )$

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\(\langle 8,42,10 \rangle\)

Question 3

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The lengths of two vectors a and b, and the angle $\theta$ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ . $| \mathbf { a } | = 0.12 , | \mathbf { b } | = 1.25 , \theta = 85 ^ { \circ }$

Given $\mathbf { u } = \langle 4 , - 3 \rangle$ and $\mathbf { v } = \langle 9,2 \rangle$ , calculate $\operatorname { proj } _ { \mathbf { v } } \mathbf { u }$ and then resolve $\mathbf { u }$ into $\mathbf { u } _ { 1 }$ and $\mathbf { u } _ { 2 }$ .

Find the area of the parallelogram determined by the given vectors. $\mathbf { u } = \langle 0 , - 2,3 \rangle , \mathbf { v } = \langle 5 , - 5,0 \rangle$

Find the area of the parallelogram determined by the given vectors. $\mathbf { u } = 2 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } , \mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }$

Describe the surface represented by the given equation. $x = 3$

Given $\mathbf { u }$ and $\mathbf { v }$ in the figure, sketch $\mathbf { v } - \mathbf { u }$ .

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $\mathbf { u } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } , \mathbf { v } = \mathbf { j } + \mathbf { k }$

Two vectors u and v are given. Find their dot product $\mathbf { u } \cdot \mathbf { v }$ . $\mathbf { u } = \langle - 3,0,3 \rangle , \mathbf { v } = \langle 2,4 , \frac { 1 } { 3 } \rangle$

A description of a line is given. Find parametric equations for the line. The line perpendicular to the xz-plane that contains the point $( 3 , - 1,4 )$ .

Find the vector v with initial point P and terminal point Q. $P ( 2 , - 1,0 )$ , $Q ( 0 , - 2,5 )$

Find the vector with initial point $P ( 6,3 )$ and terminal point $Q ( - 6 , - 3 )$ .

Describe the trace of the sphere $( x - 1 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 3 ) ^ { 2 } = 144$ through (a) the xz-plane in (b) the plane $z = - 2$ .

Find parametric equations for the line that passes through the points P and Q. $P ( 2 , - 1 , - 1 ) , \quad Q ( 0,1 , - 3 )$

Find an equation of a sphere with the given radius r and center C. $r = \sqrt { 7 }$ ; $C ( 3 , - 1,0 )$

Find $\left| \frac { 1 } { 2 } \mathbf { v } \right|$ , $| \mathbf { u } - \mathbf { v } |$ and $| \mathbf { u } | - | \mathbf { v } |$ , given that $\mathbf { u } = \langle 1$ , $\text { 3) }$ and $\mathbf { v } = \langle - 2,1 \rangle$ .

Given that the forces $\mathbf { F } _ { 1 } = \langle - 10,3 \rangle$ , $\mathbf { F } _ { 2 } = \langle - 4,1 \rangle$ and $\mathbf { F } _ { ] } = \langle 4 , - 10 \rangle$ are acting on a point $P$ , find the resultant force, magnitude and the additional force required in order for the forces to be in equilibrium.

Find $| v |$ and $| \mathbf { u } + \mathbf { v } |$ , given that $\mathbf { u } = 2 \mathbf { i } - \mathbf { j }$ and $\mathbf { v } = \mathbf { i }$ .

Find the vectors $\mathbf { u } + \mathbf { v }$ , $\mathbf { u } - \mathbf { v }$ , and $3 \mathbf { u } - \frac { 1 } { 2 } \mathbf { v }$ . $\mathbf { u } = \langle a , 2 b , 3 c \rangle , \mathbf { v } = \langle - a , b , - 2 c \rangle$