Vector Form - AP Calculus BC

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients. That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ . So for , we can derive the vector form $\left \langle -8,8,-1\right \rangle$ .

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients. That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ . So for , we can derive the vector form $\left \langle 7,2,-7\right \rangle$ .

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients. That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ . So for , we can derive the vector form $\left \langle 1,-10,0\right \rangle$ .

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients.

That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ .

So for , we can derive the vector form $\left \langle -1,9,3\right \rangle$ .

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients.

That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ .

So for , we can derive the vector form $\left \langle 1,0,-3\right \rangle$ .

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients.

That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ .

So for , we can derive the vector form $\left \langle 2,-1,0\right \rangle$ .

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Question

Assume that Billy fired himself out of a circus cannon at a velocity of at an elevation angle of degrees. Write this in vector component form.

Answer

The firing of the cannon has both x and y components.

Write the formula that distinguishes the x and y direction and substitute.

$\left \langle x,y\right \rangle= \left \langle vcos(\theta),vsin(\theta)\right \rangle$

Ensure that the calculator is in degree mode before you solve.

$\left \langle 25cos25,25sin25\right \rangle = \left \langle22.658,10.565 \right \rangle$

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Question

Express in vector form.

Answer

In order to express in vector form, we will need to map its , , and coefficients to its -, -, and -coordinates.

Thus, its vector form is

$\left \langle 0,-1,7\right \rangle$ .

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients.

That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ .

So for , we can derive the vector form $\left \langle 0,-4,-5\right \rangle$ .

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients.

That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ .

So for , we can derive the vector form $\left \langle 1,2,3\right \rangle$ .

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Question

Express in vector form.

Answer

The correct form of x,y, and z of a vector is represented in the order of i, j, and k, respectively. The coefficients of i,j, and k are used to write the vector form.

$-i-k = -1i+0j-1k = \left \langle -1,0,-1\right \rangle$

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Question

Express in vector form.

Answer

The x,y, and z of a vector is represented in the order of i, j, and k, respectively. Use the coefficients of i,j, and k to write the vector form.

$-10j=0i-10j+0k = \left \langle 0,-10,0\right \rangle$

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Question

Find the vector form of to .

Answer

When we are trying to find the vector form we need to remember the formula which states to take the difference between the ending and starting point.

Thus we would get:

Given and

$\overrightarrow{v}=[d-a, e-b, f-c]$

In our case we have ending point at and our starting point at .

Therefore we would set up the following and simplify.

$\overrightarrow{v}=[6-0,3-1,1-3]=[6,2,-2]$

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients.

That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ .

So for , we can derive the vector form $\left \langle 7,0,-7\right \rangle$ .

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Question

What is the vector form of ?

Answer

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients.

That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ .

So for , we can derive the vector form $\left \langle 0,3,5\right \rangle$ .

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Question

Find the dot product of the 2 vectors. $\left \langle a,2a\right \rangle \cdot \left \langle a,a\right \rangle$

Answer

The dot product will give a single value answer, and not a vector as a result.

To find the dot product, use the following formula:

$\left \langle x_1,x_2\right \rangle \cdot \left \langle y_1,y_2\right \rangle=(x_1)(y_1)+(x_2)(y_2)$

$\left \langle a,2a\right \rangle \cdot \left \langle a,a\right \rangle = (a)(a)+(2a)(a)=a^2+2a^2=3a^2$

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Question

Compute: $2\vec{a}-6\vec{b}$ given the following vectors. $\vec{a}= \left \langle 1,3\right \rangle$ and $\vec{b}= \left \langle 1\right \rangle$ .

Answer

The dimensions of the vectors are mismatched.

Since vector $\vec{a}$ does not have the same dimensions as $\vec{b}$ , the answer for $2\vec{a}-6\vec{b}$ cannot be solved.

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Question

Express in vector form.

Answer

In order to express in vector form, we must use the coefficients of and to represent the -, -, and -coordinates of the vector.

Therefore, its vector form is

$\left \langle -2,7,-10\right \rangle$ .

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Question

Express in vector form.

Answer

In order to express in vector form, we must use the coefficients of and to represent the -, -, and -coordinates of the vector.

Therefore, its vector form is

$\left \langle 1,0,5\right \rangle$ .

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Question

Express in vector form.

Answer

In order to express in vector form, we must use the coefficients of and to represent the -, -, and -coordinates of the vector.

Therefore, its vector form is

$\left \langle 4,-9,0\right \rangle$ .

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