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Precalculus : Parallel and Perpendicular Vectors in Two Dimensions

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Example Questions

Example Question #11 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are parallel?

Possible Answers:

$u=\frac{1}{2}i+6j$

v=i+3j

u=8i-2j

v=8i-9j

u=4i-7j

v=2i+8j

u=7i+4j

v=28i+16j

Correct answer:

u=7i+4j

v=28i+16j

Explanation:

For two vectors, and to be parallel, x=cy , for some real number .

Recall that for a vector, $v\left<a,b\right>=ai+bj$ .

The correct answer here is

$u=7i+4j=\left<7, 4\right>$

$v=28i+16j=\left<28, 16\right>$

This is the correct answer because $u=\frac{1}{4}v$ .

$\frac{1}{4}\left<28, 16\right>=\left<7, 4\right>$

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Example Question #12 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are parallel?

Possible Answers:

$u=\frac{1}{3}i+\frac{\sqrt3}{2}j$

$v=2i+3\sqrt3j$

$u=\frac{1}{2}i+\sqrt2j$

$v=4i+4\sqrt2j$

$u=2i-\sqrt3j$

$v=i+\frac{\sqrt3}{2}j$

$u=i-\sqrt2j$

$v=-4i-4\sqrt2j$

Correct answer:

$u=\frac{1}{3}i+\frac{\sqrt3}{2}j$

$v=2i+3\sqrt3j$

Explanation:

For two vectors, and to be parallel, x=cy , for some real number .

Recall that for a vector, $v\left<a,b\right>=ai+bj$ .

The correct answer here is

$u=\frac{1}{3}i+\frac{\sqrt3}{2}j=\left<\frac{1}{3}, \frac{\sqrt3}{2}\right>$

$v=2i+3\sqrt3j=\left<2, 3\sqrt3\right>$ .

Using the formula x=cy we have our to be $u=\frac{1}{6}v$ .

Applying this we find the vector that is parallel.

$\frac{1}{6}\left<2, 3\sqrt3\right>=\left<\frac{1}{3}, \frac{\sqrt3}{2}\right>$

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Example Question #11 : Determine If Two Vectors Are Parallel Or Perpendicular

Which pair of vectors represents two parallel vectors?

Possible Answers:

$\left<11,3\right>,\left<33,3\right>$

$\left<1,1\right>,\left<2,1\right>$

$\left<6,3\right>, \left<2,1\right>$

$\left<6,3\right>,\left<4,8\right>$

$\left<1,3\right>,\left<2,4\right>$

Correct answer:

$\left<6,3\right>, \left<2,1\right>$

Explanation:

Two vectors are parallel if their cross product is . This is the same thing as saying that the matrix consisting of both vectors has determinant zero.

$\\ \vector{v_1}&=\left<x_{1},y_{1}\right>\\ \vector{v_2}&=\left<x_{2},y_{2}\right>\\ M =\begin{pmatrix} x_{1}& y_{1}\\ x_{2}&y_{2} \end{pmatrix}\\det(M)=x_{1}y_{2}-x_{2}y_{1}=0\Rightarrow\vector{v_1}\parallel\vector{v_2}$

This is only true for the correct answer.

$6\cdot 1-2\cdot3=0$

In essence each vector is a scalar multiple of the other.

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Example Question #11 : Parallel And Perpendicular Vectors In Two Dimensions

Which relationship best describes the vectors $\small <3, 5>$ and $\small <7.5, -4.5>$ ?

Possible Answers:

the same direction, but different magnitudes

perpendicular

neither parallel nor perpendicular

more information needed

parallel

Correct answer:

perpendicular

Explanation:

We can discover that these vectors are perpendicular by finding the dot product:

$\small <3, 5> \cdot <7.5, -4.5> = 22.5 - 22.5 = 0$

A dot product of zero for two non zero vectors means that they are perpedicular vectors.

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Example Question #15 : Matrices And Vectors

Which relationship best describes the two vectors $\small <1.2, 0.8>$ and $\small <-9, -6>$ ?

Possible Answers:

both parallel and perpendicular

neither parallel nor perpendicular

parallel

perpendicular

more information is needed

Correct answer:

parallel

Explanation:

To show that these are parallel, we have to find their magnitudes using the Pythagorean Theorem:

$\small (1.2)^2 + (0.8)^2 = C^2$

$\small 2.08 = C^2$

$\small \sqrt{2.08}=C$

$\small (-9)^2 + (-6)^2 = C^2$

$\small 117 = C^2$

$\small \sqrt{117}=C$

To multiply $\small \sqrt{2.08}*\sqrt{117}$ , multiply the numbers within the radicals first, then take the square root:

$\small \sqrt{243.36 }= 15.6$

Now we have to find the dot product and compare it to the product of these two magnitudes. If they are the same, or if they differ only by sign [one is the negative version of the other] then the two lines are parallel.

$\small \small <1.2, 0.8> \cdot <-9, -6> = -10.8 - 4.8 = -15.6$

This is the negative version of the magnitudes' product! This means that the two vectors are parallel.

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Example Question #12 : Parallel And Perpendicular Vectors In Two Dimensions

Find the dot product of the two vectors

$<1,\,0>$

and

$<0,\,1>$ .

Possible Answers:

Correct answer:

Explanation:

To find the dot product of two vectors,

find the product of the x-components,

find the product of the y-components,

and then find the sum.

$1\cdot 0+0\cdot1=0$

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Example Question #13 : Parallel And Perpendicular Vectors In Two Dimensions

Find the dot product of the two vectors

$<a,\,b>$

and

$<c,\,d>$ .

Possible Answers:

ac+bd

$\frac{d-b}{c-a}$

abcd

ac-bd

Correct answer:

ac+bd

Explanation:

To find the dot product of two vectors,

find the product of the x-components,

find the product of the y-components,

and then find the sum.

ac+bd

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Example Question #14 : Parallel And Perpendicular Vectors In Two Dimensions

Find the dot product of the two vectors

$<1,\,1>$

and

$<5,\,5>$ .

Possible Answers:

Correct answer:

Explanation:

To find the dot product of two vectors,

find the product of the x-components,

find the product of the y-components,

and then find the sum.

$1\cdot 5+1\cdot5=5+5=10$

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Example Question #16 : Determine If Two Vectors Are Parallel Or Perpendicular

Evaluate the dot product of the following two vectors:

$\left \langle -7, 4,-3\right \rangle\cdot \left \langle 8,11,-12\right \rangle=$

Possible Answers:

Correct answer:

Explanation:

To find the dot product of two vectors, we multiply the corresponding terms of each vector and then add the results together, as expressed by the following formula:

$\left \langle a_1,a_2,a_3\right \rangle\cdot \left \langle b_1,b_2,b_3\right \rangle=a_1b_1+a_2b_2+a_3b_3$

$\left \langle -7, 4,-3\right \rangle\cdot \left \langle 8,11,-12\right \rangle=(-7)(8)+(4)(11)+(-3)(-12)=24$

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Example Question #15 : Matrices And Vectors

Let

u_1=<1,1>

u_2=<1,0>

Find the dot product of the two vectors

$u_1\bullet u_2$ .

Possible Answers:

Correct answer:

Explanation:

Let

u=<x,y>

The dot product $u_1\bullet u_2$ is equal to

$x_1x_2+y_1y_2 = 1\cdot 1+1\cdot0 =1+0=1$ .

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Precalculus : Parallel and Perpendicular Vectors in Two Dimensions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #11 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #12 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #11 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #11 : Parallel And Perpendicular Vectors In Two Dimensions

Example Question #15 : Matrices And Vectors

Example Question #12 : Parallel And Perpendicular Vectors In Two Dimensions

Example Question #13 : Parallel And Perpendicular Vectors In Two Dimensions

Example Question #14 : Parallel And Perpendicular Vectors In Two Dimensions

Example Question #16 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #15 : Matrices And Vectors

All Precalculus Resources

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