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Precalculus : Geometric Vectors

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Example Questions

Example Question #21 : Geometric Vectors

Subtract: $\left \langle3, 4 \right \rangle - \left \langle -4,-3\right \rangle$ $\displaystyle \left \langle3, 4 \right \rangle - \left \langle -4,-3\right \rangle$

Possible Answers:

$\left \langle 7,-7\right \rangle$ $\displaystyle \left \langle 7,-7\right \rangle$

$\left \langle 1,-1\right \rangle$ $\displaystyle \left \langle 1,-1\right \rangle$

$\left \langle 7,7\right \rangle$ $\displaystyle \left \langle 7,7\right \rangle$

$\left \langle -7,7\right \rangle$ $\displaystyle \left \langle -7,7\right \rangle$

$\left \langle -1,1\right \rangle$ $\displaystyle \left \langle -1,1\right \rangle$

Correct answer:

$\left \langle 7,7\right \rangle$ $\displaystyle \left \langle 7,7\right \rangle$

Explanation:

Subtract the first value of the first vector, and the second value of the first vector with the second value of the second vector.

$\left \langle3, 4 \right \rangle - \left \langle -4,-3\right \rangle = \left \langle 3-(-4),4-(-3)\right \rangle$ $\displaystyle \left \langle3, 4 \right \rangle - \left \langle -4,-3\right \rangle = \left \langle 3-(-4),4-(-3)\right \rangle$

Double negative signs are converted to a positive sign.

$\left \langle 3-(-4),4-(-3)\right \rangle = \left \langle 7,7\right \rangle$ $\displaystyle \left \langle 3-(-4),4-(-3)\right \rangle = \left \langle 7,7\right \rangle$

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Example Question #21 : Evaluate Geometric Vectors

Simplify: $\left \langle 2,-3,4\right \rangle+\left \langle-3,-3 \right \rangle-\left \langle5 \right \rangle$ $\displaystyle \left \langle 2,-3,4\right \rangle+\left \langle-3,-3 \right \rangle-\left \langle5 \right \rangle$

Possible Answers:

$\left \langle -1,0,-1\right \rangle$ $\displaystyle \left \langle -1,0,-1\right \rangle$

$\left \langle -1,-6,-1\right \rangle$ $\displaystyle \left \langle -1,-6,-1\right \rangle$

$\left \langle -8\right \rangle$ $\displaystyle \left \langle -8\right \rangle$

$\textup{ Cannot be solved.}$ $\displaystyle \textup{ Cannot be solved.}$

$\left \langle -6,-6,4\right \rangle$ $\displaystyle \left \langle -6,-6,4\right \rangle$

Correct answer:

$\textup{ Cannot be solved.}$ $\displaystyle \textup{ Cannot be solved.}$

Explanation:

The dimensions of the vectors are not the same. Placeholders cannot be added to a vector. Therefore, the values of the vectors cannot be added.

The correct answer is: $\textup{ Cannot be solved.}$ $\displaystyle \textup{ Cannot be solved.}$

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Example Question #21 : Geometric Vectors

Find the norm of the vector $\vec{v}=[2,4,3]$ $\displaystyle \vec{v}=[2,4,3]$.

Possible Answers:

$\left | \vec{v}\right |= \sqrt{29}\approx 5.385$ $\displaystyle \left | \vec{v}\right |= \sqrt{29}\approx 5.385$

$\left | \vec{v}\right |= 2\sqrt{6}\approx 4.899$ $\displaystyle \left | \vec{v}\right |= 2\sqrt{6}\approx 4.899$

$\left | \vec{v}\right |= \sqrt{9} = 3$ $\displaystyle \left | \vec{v}\right |= \sqrt{9} = 3$

$\left | \vec{v}\right |= 9$ $\displaystyle \left | \vec{v}\right |= 9$

Correct answer:

$\left | \vec{v}\right |= \sqrt{29}\approx 5.385$ $\displaystyle \left | \vec{v}\right |= \sqrt{29}\approx 5.385$

Explanation:

We find the norm of a vector by finding the sum of each element squared and then taking the square root.

$\left | \vec{v} \right |= \sqrt{2^2+4^2+3^2}=\sqrt{29}$ $\displaystyle \left | \vec{v} \right |= \sqrt{2^2+4^2+3^2}=\sqrt{29}$.

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Example Question #22 : Geometric Vectors

Find the norm of the vector $\vec{r}=7\hat{i} +4 \hat{j}+5 \hat{k}$ $\displaystyle \vec{r}=7\hat{i} +4 \hat{j}+5 \hat{k}$.

Possible Answers:

$\left | \vec{r}\right | = 4$ $\displaystyle \left | \vec{r}\right | = 4$

$\left | \vec{r}\right | = 2 \sqrt{13}$ $\displaystyle \left | \vec{r}\right | = 2 \sqrt{13}$

$\left | \vec{r} \ \right | = 3 \sqrt{10}$ $\displaystyle \left | \vec{r} \ \right | = 3 \sqrt{10}$

$\left | \vec{r}\right | = 16$ $\displaystyle \left | \vec{r}\right | = 16$

Correct answer:

$\left | \vec{r} \ \right | = 3 \sqrt{10}$ $\displaystyle \left | \vec{r} \ \right | = 3 \sqrt{10}$

Explanation:

We find the norm of a vector by finding the sum of each component squared and then taking the square root of that sum.

$\left | \vec{r}\right | = \sqrt{ 7^2+4^2+5^2 } = \sqrt{ 90 } = 3 \sqrt{10}$ $\displaystyle \left | \vec{r}\right | = \sqrt{ 7^2+4^2+5^2 } = \sqrt{ 90 } = 3 \sqrt{10}$

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Example Question #23 : Geometric Vectors

Find the norm of the vector: <3,-12,5.6,2,-9> $\displaystyle < 3,-12,5.6,2,-9>$

Possible Answers:

$\sqrt{184.36}\approx 13.58$ $\displaystyle \sqrt{184.36}\approx 13.58$

$\sqrt{31.6}\approx 5.62$ $\displaystyle \sqrt{31.6}\approx 5.62$

$\sqrt{269.36}\approx 16.41$ $\displaystyle \sqrt{269.36}\approx 16.41$

31.6 $\displaystyle 31.6$

269.36 $\displaystyle 269.36$

Correct answer:

$\sqrt{269.36}\approx 16.41$ $\displaystyle \sqrt{269.36}\approx 16.41$

Explanation:

The norm of a vector is also known as the length of the vector. The norm is given by the formula:

$\sqrt{\sum_{i=1}^nv_i^2}=\left \| v\right \|$ $\displaystyle \sqrt{\sum_{i=1}^nv_i^2}=\left \| v\right \|$.

Here, we have

$\left \|v \right \|=\sqrt{3^2+(-12)^2+5.6^2+2^2+(-9)^2}=\sqrt{269.36}$ $\displaystyle \left \|v \right \|=\sqrt{3^2+(-12)^2+5.6^2+2^2+(-9)^2}=\sqrt{269.36}$,

the correct answer.

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Example Question #23 : Geometric Vectors

Find the norm of vector $\left \langle -2,2\right \rangle$ $\displaystyle \left \langle -2,2\right \rangle$.

Possible Answers:

$\displaystyle 4$

$\displaystyle 0$

$\frac{1}{4}$ $\displaystyle \frac{1}{4}$

$2\sqrt2$ $\displaystyle 2\sqrt2$

$\displaystyle 2$

Correct answer:

$2\sqrt2$ $\displaystyle 2\sqrt2$

Explanation:

Write the formula to find the norm, or the length the vector.

$\left \| v\right \|=\sqrt{a_{1}^2+b_{2}^2}$ $\displaystyle \left \| v\right \|=\sqrt{a_{1}^2+b_{2}^2}$

Substitute the known values of the vector and solve.

$\left \| v\right \|=\sqrt{(-2)^2+2^2}= \sqrt{4+4}=\sqrt8=2\sqrt2$ $\displaystyle \left \| v\right \|=\sqrt{(-2)^2+2^2}= \sqrt{4+4}=\sqrt8=2\sqrt2$

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Example Question #26 : Evaluate Geometric Vectors

Find the norm (magnitude) of the following vector:

$\vec{b}=3i+5j-6k$ $\displaystyle \vec{b}=3i+5j-6k$

Possible Answers:

$\displaystyle 70$

$\sqrt{72}$ $\displaystyle \sqrt{72}$

$\sqrt{70}$ $\displaystyle \sqrt{70}$

$-\sqrt{70}$ $\displaystyle -\sqrt{70}$

$\displaystyle 36$

Correct answer:

$\sqrt{70}$ $\displaystyle \sqrt{70}$

Explanation:

Use the following equation to find the magnitude of a vector:

$\left \| \vec{b}\right \|=\sqrt{i^2+j^2+k^2}$ $\displaystyle \left \| \vec{b}\right \|=\sqrt{i^2+j^2+k^2}$

In this case we have:

$\vec{b}=3i+5j-6k$ $\displaystyle \vec{b}=3i+5j-6k$

So plug in our values:

$\left \| \vec{b}\right \|=\sqrt{3^2+5^2+(-6)^2}=\sqrt{70}$ $\displaystyle \left \| \vec{b}\right \|=\sqrt{3^2+5^2+(-6)^2}=\sqrt{70}$

So:

$\left \| \vec{b} \right \|=\sqrt{70}$ $\displaystyle \left \| \vec{b} \right \|=\sqrt{70}$

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Example Question #24 : Geometric Vectors

Find the product of the vector $\vec{r}= 11\hat{i}+7\hat{j}$ $\displaystyle \vec{r}= 11\hat{i}+7\hat{j}$ and the scalar a=4 $\displaystyle a=4$.

Possible Answers:

$a\vec{r}=72$ $\displaystyle a\vec{r}=72$

$a\vec{r}=44 \hat{j}+28\vec{i}$ $\displaystyle a\vec{r}=44 \hat{j}+28\vec{i}$

$a\vec{r}=44 \hat{i}+28\vec{j}$ $\displaystyle a\vec{r}=44 \hat{i}+28\vec{j}$

$a\vec{r}=72 \hat{i}$ $\displaystyle a\vec{r}=72 \hat{i}$

Correct answer:

$a\vec{r}=44 \hat{i}+28\vec{j}$ $\displaystyle a\vec{r}=44 \hat{i}+28\vec{j}$

Explanation:

When multiplying a vector by a scalar we multiply each component of the vector by the scalar and the result is a vector:

$a \vec{r}= 4 \cdot (11 \hat{i} + 7 \hat{j}) = 44 \hat{i}+28 \hat{j}$ $\displaystyle a \vec{r}= 4 \cdot (11 \hat{i} + 7 \hat{j}) = 44 \hat{i}+28 \hat{j}$

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Precalculus : Geometric Vectors

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #21 : Geometric Vectors

Example Question #21 : Evaluate Geometric Vectors

Example Question #21 : Geometric Vectors

Example Question #22 : Geometric Vectors

Example Question #23 : Geometric Vectors

Example Question #23 : Geometric Vectors

Example Question #26 : Evaluate Geometric Vectors

Example Question #24 : Geometric Vectors

All Precalculus Resources

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