QuestionFind the unit vector in the direction of the sum of the vectors $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$ .

Studdy Solution

STEP 1

Assumptions1. The vectors are given as $\vec{a}= \hat{i}-\hat{j}+ \hat{k}$ $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$ . We need to find the unit vector in the direction of the sum of these vectors.

STEP 2

First, we need to find the sum of the vectors. We can do this by adding the corresponding components of the vectors.
$\vec{a} + \vec{b} = (2 \hat{i}-\hat{j}+2 \hat{k}) + (-\hat{i}+\hat{j}+ \hat{k})$

STEP 3

implify the above expression to find the sum of the vectors.
$\vec{a} + \vec{b} = (2 \hat{i} - \hat{i}) + (-\hat{j} + \hat{j}) + (2 \hat{k} +3 \hat{k})$

STEP 4

Calculate the sum of the vectors.
$\vec{a} + \vec{b} = \hat{i} + \hat{k}$

STEP 5

Now that we have the sum of the vectors, we need to find the magnitude of this vector. The magnitude of a vector is given by the square root of the sum of the squares of its components.
$|\vec{a} + \vec{b}| = \sqrt{(\hat{i})^2 + (5 \hat{k})^2}$

STEP 6

Calculate the magnitude of the vector.
$|\vec{a} + \vec{b}| = \sqrt{1^2 +5^2}$

STEP 7

implify the above expression to find the magnitude of the vector.
$|\vec{a} + \vec{b}| = \sqrt{1 +25}$

STEP 8

Calculate the magnitude of the vector.
$|\vec{a} + \vec{b}| = \sqrt{26}$

STEP 9

Now that we have the magnitude of the vector, we can find the unit vector in the direction of the sum of the vectors. The unit vector is given by the vector divided by its magnitude.
$\hat{u} = \frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|}$

STEP 10

Plug in the values for the vector and its magnitude to calculate the unit vector.
$\hat{u} = \frac{\hat{i} +5 \hat{k}}{\sqrt{26}}$ The unit vector in the direction of the sum of the vectors is $\frac{\hat{i} +5 \hat{k}}{\sqrt{26}}$ .

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QuestionFind the unit vector in the direction of the sum of the vectors $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$ .

Studdy Solution

STEP 1

Assumptions1. The vectors are given as $\vec{a}= \hat{i}-\hat{j}+ \hat{k}$ $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$ . We need to find the unit vector in the direction of the sum of these vectors.

STEP 2

First, we need to find the sum of the vectors. We can do this by adding the corresponding components of the vectors.
$\vec{a} + \vec{b} = (2 \hat{i}-\hat{j}+2 \hat{k}) + (-\hat{i}+\hat{j}+ \hat{k})$

STEP 3

implify the above expression to find the sum of the vectors.
$\vec{a} + \vec{b} = (2 \hat{i} - \hat{i}) + (-\hat{j} + \hat{j}) + (2 \hat{k} +3 \hat{k})$

STEP 4

Calculate the sum of the vectors.
$\vec{a} + \vec{b} = \hat{i} + \hat{k}$

STEP 5

Now that we have the sum of the vectors, we need to find the magnitude of this vector. The magnitude of a vector is given by the square root of the sum of the squares of its components.
$|\vec{a} + \vec{b}| = \sqrt{(\hat{i})^2 + (5 \hat{k})^2}$

STEP 6

Calculate the magnitude of the vector.
$|\vec{a} + \vec{b}| = \sqrt{1^2 +5^2}$

STEP 7

implify the above expression to find the magnitude of the vector.
$|\vec{a} + \vec{b}| = \sqrt{1 +25}$

STEP 8

Calculate the magnitude of the vector.
$|\vec{a} + \vec{b}| = \sqrt{26}$

STEP 9

Now that we have the magnitude of the vector, we can find the unit vector in the direction of the sum of the vectors. The unit vector is given by the vector divided by its magnitude.
$\hat{u} = \frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|}$

STEP 10

Plug in the values for the vector and its magnitude to calculate the unit vector.
$\hat{u} = \frac{\hat{i} +5 \hat{k}}{\sqrt{26}}$ The unit vector in the direction of the sum of the vectors is $\frac{\hat{i} +5 \hat{k}}{\sqrt{26}}$ .

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QuestionFind the unit vector in the direction of the sum of the vectors a⃗=2i^−j^+2k^\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}a=2i^−j^​+2k^ and b⃗=−i^+j^+3k^\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}b=−i^+j^​+3k^.

Studdy Solution

STEP 1

Assumptions1. The vectors are given as a⃗=i^−j^+k^\vec{a}= \hat{i}-\hat{j}+ \hat{k}a=i^−j^​+k^ b⃗=−i^+j^+3k^\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}b=−i^+j^​+3k^. We need to find the unit vector in the direction of the sum of these vectors.

STEP 2

First, we need to find the sum of the vectors. We can do this by adding the corresponding components of the vectors.a⃗+b⃗=(2i^−j^+2k^)+(−i^+j^+k^)\vec{a} + \vec{b} = (2 \hat{i}-\hat{j}+2 \hat{k}) + (-\hat{i}+\hat{j}+ \hat{k})a+b=(2i^−j^​+2k^)+(−i^+j^​+k^)

STEP 3

implify the above expression to find the sum of the vectors.a⃗+b⃗=(2i^−i^)+(−j^+j^)+(2k^+3k^)\vec{a} + \vec{b} = (2 \hat{i} - \hat{i}) + (-\hat{j} + \hat{j}) + (2 \hat{k} +3 \hat{k})a+b=(2i^−i^)+(−j^​+j^​)+(2k^+3k^)

STEP 4

Calculate the sum of the vectors.a⃗+b⃗=i^+k^\vec{a} + \vec{b} = \hat{i} + \hat{k}a+b=i^+k^

STEP 5

Now that we have the sum of the vectors, we need to find the magnitude of this vector. The magnitude of a vector is given by the square root of the sum of the squares of its components.∣a⃗+b⃗∣=(i^)2+(5k^)2|\vec{a} + \vec{b}| = \sqrt{(\hat{i})^2 + (5 \hat{k})^2}∣a+b∣=(i^)2+(5k^)2​

STEP 6

Calculate the magnitude of the vector.∣a⃗+b⃗∣=12+52|\vec{a} + \vec{b}| = \sqrt{1^2 +5^2}∣a+b∣=12+52​

STEP 7

implify the above expression to find the magnitude of the vector.∣a⃗+b⃗∣=1+25|\vec{a} + \vec{b}| = \sqrt{1 +25}∣a+b∣=1+25​

STEP 8

Calculate the magnitude of the vector.∣a⃗+b⃗∣=26|\vec{a} + \vec{b}| = \sqrt{26}∣a+b∣=26​

STEP 9

Now that we have the magnitude of the vector, we can find the unit vector in the direction of the sum of the vectors. The unit vector is given by the vector divided by its magnitude.u^=a⃗+b⃗∣a⃗+b⃗∣\hat{u} = \frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|}u^=∣a+b∣a+b​

STEP 10

Plug in the values for the vector and its magnitude to calculate the unit vector.u^=i^+5k^26\hat{u} = \frac{\hat{i} +5 \hat{k}}{\sqrt{26}}u^=26​i^+5k^​The unit vector in the direction of the sum of the vectors is i^+5k^26\frac{\hat{i} +5 \hat{k}}{\sqrt{26}}26​i^+5k^​.

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QuestionFind the unit vector in the direction of the sum of the vectors $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$ .

Assumptions1. The vectors are given as $\vec{a}= \hat{i}-\hat{j}+ \hat{k}$ $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$ . We need to find the unit vector in the direction of the sum of these vectors.

First, we need to find the sum of the vectors. We can do this by adding the corresponding components of the vectors.
$\vec{a} + \vec{b} = (2 \hat{i}-\hat{j}+2 \hat{k}) + (-\hat{i}+\hat{j}+ \hat{k})$

implify the above expression to find the sum of the vectors.
$\vec{a} + \vec{b} = (2 \hat{i} - \hat{i}) + (-\hat{j} + \hat{j}) + (2 \hat{k} +3 \hat{k})$

Calculate the sum of the vectors.
$\vec{a} + \vec{b} = \hat{i} + \hat{k}$

Now that we have the sum of the vectors, we need to find the magnitude of this vector. The magnitude of a vector is given by the square root of the sum of the squares of its components.
$|\vec{a} + \vec{b}| = \sqrt{(\hat{i})^2 + (5 \hat{k})^2}$

Calculate the magnitude of the vector.
$|\vec{a} + \vec{b}| = \sqrt{1^2 +5^2}$

implify the above expression to find the magnitude of the vector.
$|\vec{a} + \vec{b}| = \sqrt{1 +25}$

Calculate the magnitude of the vector.
$|\vec{a} + \vec{b}| = \sqrt{26}$

Now that we have the magnitude of the vector, we can find the unit vector in the direction of the sum of the vectors. The unit vector is given by the vector divided by its magnitude.
$\hat{u} = \frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|}$

Plug in the values for the vector and its magnitude to calculate the unit vector.
$\hat{u} = \frac{\hat{i} +5 \hat{k}}{\sqrt{26}}$ The unit vector in the direction of the sum of the vectors is $\frac{\hat{i} +5 \hat{k}}{\sqrt{26}}$ .