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Precalculus : Pre-Calculus

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Example Questions

Example Question #3 : Graphs Of Polar Equations

Please choose the best answer from the following choices.

Determine the equation of the following polar graph:

Msp6661d6g1fca347a4f9a00002g2fc12725bd10gh

Possible Answers:

$r=3cos(2\theta )$

$r=3sin(2\theta )$

$r=cos(2\theta )$

$r=sin(2\theta )$

Correct answer:

$r=3cos(2\theta )$

Explanation:

Each loop is on the line of each axis, which means the equation will have a cosine.

The loops stretch out to the third unit on each axis, which means there will be a three in front of the cosine.

Lastly, there are 4 loops. Whenever there is an even number of loops, you divide that number by 2, and that is the number which goes in front of theta in the parentheses. In this case, it is $2\theta$ because 4/2=2 .

Thus our answer is,

$r=3cos(2\theta )$ .

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Example Question #1 : Graphs Of Polar Equations

Which of the following polar equations would produce this graph?

Polar

Possible Answers:

$r=2+2cos(\theta )$

$r=sin(\theta )$

$r=1+1sin(\theta)$

$r=2+2sin(\theta )$

$r=1+1cos(\theta)$

Correct answer:

$r=1+1cos(\theta)$

Explanation:

You can tell from the direction of the cardioid, extending horizontally, that you are dealing with a cosine graph. Then you should note that the y-intercepts are at 1 and -1, telling you that the first number must be 1. And since the graph extends to x = 2, you should note that the two numeric terms in $r=a+bcos(\theta )$ should add to 2, making the correct choice $r=1+1cos(\theta )$ .

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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:

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

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

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

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

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

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

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

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

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:

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

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

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

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

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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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #3 : Graphs Of Polar Equations

Example Question #1 : Graphs Of Polar Equations

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

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