AP Calculus BC - Vectors and Vector Operations

Compute the following: $\left \langle 1,3,2\right \rangle-\left \langle 2,-4,1\right \rangle$ .

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

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

$\left \langle -2,0,1\right \rangle$

$\left \langle 3,-7,2\right \rangle$

Explanation

The formula for subtracting vectors is $a-b=\left \langle a_1-b_!,a_2-b_2,a_3-b_3\right \rangle$ . Plugging in the values we were given, we get $\left \langle (1-2),(3--4),(2-1)\right \rangle=\left \langle -1,7,1\right \rangle$

Find the dot product of the vectors $\left \langle 8,9,5\right \rangle$ and $\left \langle 0,2,-7\right \rangle$

Explanation

To find the dot product of two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we use the formula $a\cdot b=a_1b_1+a_2b_2+a_3b_3$

Using the vectors from the problem statement, we get

Find the dot product between the vectors $\left \langle -7,-1,0\right \rangle$ and $\left \langle 2,3,0\right \rangle$

Explanation

To find the dot product between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we apply the following formula:
$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we get

Determine whether the two vectors, $\overrightarrow{a}=\begin{bmatrix}2 \0 \5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}10 \7 \-2 \end{bmatrix}$ , are orthogonal or not.

The two vectors are not orthogonal.

The two vectors are orthogonal.

Explanation

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

$\overrightarrow{a}\cdot\overrightarrow{b}=0\rightarrow \overrightarrow{a}\perp\overrightarrow{b}$

To find the dot product of two vectors given the notation

$\overrightarrow{a}\begin{bmatrix} a_1\a_2 \ a_3 \end{bmatrix};\overrightarrow{b}\begin{bmatrix} b_1\b_2 \ b_3 \end{bmatrix}$

Simply multiply terms across rows:

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3$

For our vectors, $\overrightarrow{a}=\begin{bmatrix}2 \0 \5 \end{bmatrix}$ and $\overrightarrow{b}=\begin{bmatrix}10 \7 \-2 \end{bmatrix}$

$\overrightarrow{a}\cdot\overrightarrow{b}=20+0-10=10$

The two vectors are not orthogonal.

Find the dot product between the vectors $\left \langle 2,y,7\right \rangle$ and $\left \langle 3,4,-1\right \rangle$

Explanation

To find the dot product between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we apply the following formula:

$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we get

Find the dot product of the vectors $\left \langle 2,-4,1\right \rangle$ and $\left \langle -1,3,6\right \rangle$

Explanation

To find the dot product of two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we apply the formula $a\cdot b=a_1b_1+a_2b_2+a_3b_3$

Using the vectors from the problem statement, we get

Given the following two vectors, $\overrightarrow{a}=1\widehat{i}+11\widehat{j}$ and $\overrightarrow{b}=15\widehat{i}-2\widehat{j}$ , calculate the dot product between them, $\overrightarrow{a}\cdot\overrightarrow{b}$ .

Explanation

The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.

$\overrightarrow{a}=a_1\widehat{x_1}+a_2\widehat{x_2}+a_3\widehat{x_3}+...+a_n\widehat{x_n}$

$\overrightarrow{b}=b_1\widehat{x_1}+b_2\widehat{x_2}+b_3\widehat{x_3}+...+b_n\widehat{x_n}$

$\overrightarrow{a}\cdot\overrightarrow{b}=a_1b_1+a_2b_2+a_3b_3+...+a_nb_n$

Note that the dot product is a scalar value rather than a vector; there's no directional term.

Now considering our problem, we're given the vectors $\overrightarrow{a}=1\widehat{i}+11\widehat{j}$ and $\overrightarrow{b}=15\widehat{i}-2\widehat{j}$

The dot product can be found following the example above:

$\overrightarrow{a}\cdot\overrightarrow{b}=(15)+(-22)=-7$

Find the dot product between the vectors $\left \langle 2,7,20\right \rangle$ and $\left \langle 5,6,1\right \rangle$

Explanation

To find the dot product between the vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula $a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we then get

Find the dot product between the vectors $\left \langle 3x,y,5z\right \rangle$ and $\left \langle x^2,y^3,z^4\right \rangle$

Explanation

To find the dot product between the vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula