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Precalculus : Matrices and Vectors

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Example Questions

Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are perpendicular?

Possible Answers:

$u=\left<4, -5\right>$

$v=\left<2, 2\right>$

$u=\left<1, 9\right>$

$v=\left<-9, -2\right>$

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

$v=\left<0, -4\right>$

$u=\left<3, 4\right>$

$v=\left<-8, 6\right>$

Correct answer:

$u=\left<3, 4\right>$

$v=\left<-8, 6\right>$

Explanation:

Two vectors are perpendicular when their dot product equals to .

Recall how to find the dot product of two vectors $\left<v_1, v_2\right>$ and $\left<w_1, w_2\right>$

$\left<v_1, v_2\right>\cdot \left<w_1, w_2\right>=v_1w_1+v_2w_2$

The correct choice is

$u=\left<3, 4\right>$

$v=\left<-8, 6\right>$

$\left<3, 4\right>\cdot \left<-8, 6\right>=(3\times(-8))+4 \times 6=-24+24=0$

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Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are perpendicular?

Possible Answers:

$a=\left<5, 10\right>, b=\left<6, -3\right>$

$a=\left<4, 2\right>, b=\left<2, 1\right>$

$a=\left<-4, 5\right>, b=\left<6, 2\right>$

$a=\left<6, -3\right>, b=\left<1, -1\right>$

Correct answer:

$a=\left<5, 10\right>, b=\left<6, -3\right>$

Explanation:

Two vectors are perpendicular when their dot product equals to .

Recall how to find the dot product of two vectors $\left<v_1, v_2\right>$ and <w_1, w_2>

$\left<v_1, v_2\right>\cdot \left<w_1, w_2\right>=v_1w_1+v_2w_2$ .

The correct choice is,

$a=\left<5, 10\right>,b=\left<6, -3\right>$ .

$\left<5, 10\right>\cdot \left<6, -3\right>=(5)(6)+(10)(-3)=30-30=0$

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Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are perpendicular?

Possible Answers:

$c=\left<2, 1\right>,d=\left<3, -6\right>$

$c=\left<7, 8\right>,d=\left<-2, -6\right>$

$c=\left<5, -2\right>,d=\left<2, -5\right>$

$c=\left<2, 1\right>,d=\left<4, -5\right>$

Correct answer:

$c=\left<2, 1\right>,d=\left<3, -6\right>$

Explanation:

Two vectors are perpendicular when their dot product equals to .

Recall how to find the dot product of two vectors $\left<v_1, v_2\right>$ and $\left<w_1, w_2\right>$

$\left<v_1, v_2\right>\cdot \left<w_1, w_2\right>=v_1w_1+v_2w_2$ .

The correct choice is $c=\left<2, 1\right>, d=\left<3, -6\right>$ .

$\left<2, 1\right>\cdot \left<3, -6\right>=(2)(3)+(1)(-6)=6-6=0$

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Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are perpendicular?

Possible Answers:

$e=\sqrt3i+\sqrt3j$

$f=\sqrt3i+9j$

$e=3\sqrt3i-j$

$f=\sqrt3i+9j$

$e=\sqrt2i-2j$

f=2i+j

e=3i-j

$f=\sqrt3i+9j$

Correct answer:

$e=3\sqrt3i-j$

$f=\sqrt3i+9j$

Explanation:

Two vectors are perpendicular when their dot product equals to .

Recall how to find the dot product of two vectors $\left<v_1, v_2\right>$ and $\left<w_1, w_2\right>$

$\left<v_1, v_2\right>\cdot \left<w_1, w_2\right>=v_1w_1+v_2w_2$ .

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

The correct answer is then,

$e=3\sqrt3i-j=\left<3\sqrt3, -1\right>$

$f=\sqrt3i+9j=\left<\sqrt3, 9\right>$

$\left<3\sqrt3, -1\right>\cdot\left<\sqrt3, -9\right>=(3\sqrt3)(\sqrt3)+(-1)(9)=9-9=0$

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Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following vectors are perpendicular?

Possible Answers:

$g=\left<\frac{\sqrt2}{3}, 4\right>, h=\left<-\sqrt2, \frac{1}{5}\right>$

$g=\left<\sqrt2, 6\right>, h=\left<-\sqrt2, \frac{1}{6}\right>$

$g=\left<\frac{\sqrt2}{4}, 6\right>, h=\left<-\sqrt2, \frac{1}{2}\right>$

$g=\left<\frac{\sqrt2}{2}, 6\right>,h=\left<-\sqrt2, \frac{1}{6}\right>$

Correct answer:

$g=\left<\frac{\sqrt2}{2}, 6\right>,h=\left<-\sqrt2, \frac{1}{6}\right>$

Explanation:

Two vectors are perpendicular when their dot product equals to .

Recall how to find the dot product of two vectors $\left<v_1, v_2\right>$ and $\left<w_1, w_2\right>$

$\left<v_1, v_2\right>\cdot \left<w_1, w_2\right>=v_1w_1+v_2w_2$ .

The correct answer is then,

$g=\left<\frac{\sqrt2}{2}, 6\right>, h=\left<-\sqrt2, \frac{1}{6}\right>$

$\left<\frac{\sqrt2}{2}, 6\right>\cdot\left<-\sqrt2, \frac{1}{6}\right>=\left(\frac{\sqrt2}{2}\right)(-\sqrt2)+(6)\left(\frac{1}{6}\right)=-1+1=0$

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Example Question #2 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are perpendicular?

Possible Answers:

k=2i-3j

l=i+2j

k=-2i-3j

$l=\frac{1}{4}i+9j$

$k=\frac{1}{4}i+4j$

$l=8i+\frac{1}{2}j$

$k=\frac{1}{2}i-5j$

l=10i+j

Correct answer:

$k=\frac{1}{2}i-5j$

l=10i+j

Explanation:

Two vectors are perpendicular when their dot product equals to .

Recall how to find the dot product of two vectors $\left<v_1, v_2\right>$ and $\left<w_1, w_2\right>$

$\left<v_1, v_2\right>\cdot\left<w_1, w_2\right>=v_1w_1+v_2w_2$ .

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

The correct answer is then,

$k=\frac{1}{2}i-5j=\left<\frac{1}{2}, -5\right>$

$l=10i+j=\left<10, 1\right>$

$\left<\frac{1}{2}, -5\right>\cdo\left<10, 1\right>=\left(\frac{1}{2}\right)(10)+(-5)(-1)=5-5=0$

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Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are parallel?

Possible Answers:

$m=\left<-1, 5\right>, n=\left<-3, -15\right>$

$m=\left<1, 5\right>, n=\left<3, 15\right>$

$m=\left<1, 5\right>, n=\left<4, 24\right>$

$m=\left<1, 5\right>, n=\left<6, 36\right>$

Correct answer:

$m=\left<1, 5\right>, n=\left<3, 15\right>$

Explanation:

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

The correct answer here is $m=\left<1, 5\right>, n=\left<3, 15\right>$ because

n=3m .

Therefore,

$3\left<1, 5\right>=\left<3, 15\right>$ .

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Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are parallel?

Possible Answers:

$o=\left<\frac{1}{3}, 4\right>, p=\left<2, 15\right>$

$o=\left<\frac{1}{10}, 6\right>, p=\left<4, 10\right>$

$o=\left<-\frac{1}{2}, -4\right>, p=\left<-2, 16\right>$

$o=\left<\frac{1}{2}, 4\right>, p=\left<2, 16\right>$

Correct answer:

$o=\left<\frac{1}{2}, 4\right>, p=\left<2, 16\right>$

Explanation:

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

The correct answer here is $o=\left<\frac{1}{2}, 4\right>, p=\left<2, 16\right>$ because

p=4o .

Thus making the vector parallel,

$4\left<\frac{1}{2}, 4\right>=\left<2, 16\right>$ .

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Example Question #2 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are parallel?

Possible Answers:

$r=\left<\frac{\sqrt3}{2}, \frac{1}{2}\right>, s=\left<2\sqrt3, 1\right>$

$r=\left<-\frac{\sqrt3}{2}, -\frac{1}{2}\right>, s=\left<\sqrt3,\frac{1}{2}\right>$

$r=\left<\frac{\sqrt3}{2}, \frac{1}{2}\right>, s=\left<\sqrt3, 1\right>$

$r=\left<\frac{\sqrt3}{4}, \frac{1}{2}\right>, s=\left<3\sqrt3, 3\right>$

Correct answer:

$r=\left<\frac{\sqrt3}{2}, \frac{1}{2}\right>, s=\left<\sqrt3, 1\right>$

Explanation:

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

The correct answer here is $r=\left<\frac{\sqrt3}{2}, \frac{1}{2}\right>, s=\left<\sqrt3, 1\right>$ because

$r=\frac{1}{2}s$ .

Thus making the vector parallel.

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

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Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are parallel?

Possible Answers:

$u=\left<-9, 18\right>, v=\left<-3, 6\right>$

$u=\left<-9, 18\right>, v=\left<3, 6\right>$

$u=\left<\frac{7}{2}, 18\right>, v=\left<-7, 6\right>$

$u=\left<-10, 4\right>, v=\left<-2, 5\right>$

Correct answer:

$u=\left<-9, 18\right>, v=\left<-3, 6\right>$

Explanation:

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

The correct answer here is $u=\left<-9, 18\right>, v=\left<-3, 6\right>$ because

u=-3v

Making the vector parallel as follows.

$3\left<-3, 6\right>=\left<-9, 18\right>$

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Precalculus : Matrices and Vectors

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #2 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #2 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

All Precalculus Resources

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