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Precalculus : Matrices and Vectors

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Example Questions

Example Question #21 : Evaluate Geometric Vectors

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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

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Example Question #22 : Evaluate Geometric Vectors

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

Possible Answers:

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

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

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

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

Correct answer:

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

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.

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

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Example Question #23 : Evaluate Geometric Vectors

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

Possible Answers:

$\sqrt{269.36}\approx 16.41$

$\sqrt{31.6}\approx 5.62$

$\sqrt{184.36}\approx 13.58$

31.6

269.36

Correct answer:

$\sqrt{269.36}\approx 16.41$

Explanation:

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

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

Here, we have

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

the correct answer.

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Example Question #24 : Evaluate Geometric Vectors

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

Possible Answers:

$\frac{1}{4}$

$2\sqrt2$

Correct answer:

$2\sqrt2$

Explanation:

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

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

Substitute the known values of the vector and solve.

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

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Example Question #25 : Evaluate Geometric Vectors

Find the norm (magnitude) of the following vector:

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

Possible Answers:

$\sqrt{72}$

$\sqrt{70}$

$-\sqrt{70}$

Correct answer:

$\sqrt{70}$

Explanation:

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

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

In this case we have:

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

So plug in our values:

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

So:

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

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Example Question #26 : Evaluate Geometric Vectors

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

Possible Answers:

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

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

$a\vec{r}=72$

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

Correct answer:

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

Explanation:

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

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

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Precalculus : Matrices and Vectors

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #21 : Evaluate Geometric Vectors

Example Question #22 : Evaluate Geometric Vectors

Example Question #23 : Evaluate Geometric Vectors

Example Question #24 : Evaluate Geometric Vectors

Example Question #25 : Evaluate Geometric Vectors

Example Question #26 : Evaluate Geometric Vectors

All Precalculus Resources

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