Calculus 3 : Dot Product

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #131 : Dot Product

Find the dot product between \(\displaystyle \left \langle 3,2y,z^5\right \rangle\) and \(\displaystyle \left \langle 3x,yz,z^2\right \rangle\)

Possible Answers:

\(\displaystyle 9x+2x^2y+z^7\)

\(\displaystyle 9x+2yz+z^7\)

\(\displaystyle 9x+2y^2z+z^7\)

\(\displaystyle 2x^2+2y^2x-z\)

Correct answer:

\(\displaystyle 9x+2y^2z+z^7\)

Explanation:

To find the dot product between two vectors \(\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle\) and \(\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle\) we use the formula \(\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)\). Using the vectors in the problem statement, we get \(\displaystyle (3*3x)+(2y*yz)+(z^5*z^2)=9x+2y^2z+z^7\)

Example Question #132 : Dot Product

Find the dot product between \(\displaystyle \left \langle 3x,4y,z\right \rangle\) and \(\displaystyle \left \langle 4,2,7\right \rangle\)

Possible Answers:

\(\displaystyle 2x+y-9z\)

\(\displaystyle 12x+8y+7z\)

\(\displaystyle 4x-3y+5z\)

\(\displaystyle x+4y-3z\)

Correct answer:

\(\displaystyle 12x+8y+7z\)

Explanation:

To find the dot product between two vectors \(\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle\) and \(\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle\) we use the formula \(\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)\). Using the vectors in the problem statement, we get \(\displaystyle (3x*4)+(4y*2)+(z*7)=12x+8y+7z\)

Example Question #131 : Dot Product

Find the dot product between \(\displaystyle \left \langle 2,-3,4\right \rangle\) and \(\displaystyle \left \langle 5,-4,1\right \rangle\)

Possible Answers:

\(\displaystyle 20\)

\(\displaystyle 22\)

\(\displaystyle 30\)

\(\displaystyle 26\)

Correct answer:

\(\displaystyle 26\)

Explanation:

To find the dot product between two vectors \(\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle\) and \(\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle\) we use the formula \(\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)\). Using the vectors in the problem statement, we get \(\displaystyle (2*5)+(-3*-4)+(4*1)=10+12+4=26\)

Example Question #134 : Dot Product

Find the dot product between \(\displaystyle \left \langle x,-3,4z\right \rangle\) and \(\displaystyle \left \langle x,y,z\right \rangle\)

Possible Answers:

\(\displaystyle x^2+6y+4z^2\)

\(\displaystyle x^3+y+7z^2\)

\(\displaystyle x^2-5y+5z^2\)

\(\displaystyle x^2-3y+4z^2\)

Correct answer:

\(\displaystyle x^2-3y+4z^2\)

Explanation:

To find the dot product between two vectors \(\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle\) and \(\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle\) we use the formula \(\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)\). Using the vectors in the problem statement, we get \(\displaystyle (x*x)+(-3*y)+(4z*z)=x^2-3y+4z^2\)

Example Question #131 : Dot Product

Consider the vector \(\displaystyle \mathbf{v}=\mathbf{i}+7\mathbf{j}+2\mathbf{k}\).

Which of the following vectors are orthogonal to v?

Possible Answers:

\(\displaystyle 4\mathbf{i}-2\mathbf{j}+\mathbf{k}\)

\(\displaystyle 6\mathbf{i}+-3\mathbf{j}+5\mathbf{k}\)

\(\displaystyle -4\mathbf{i}+3\mathbf{j}-2\mathbf{k}\)

\(\displaystyle -5\mathbf{i}+2\mathbf{j}+3\mathbf{k}\)

\(\displaystyle -\mathbf{i}+3\mathbf{k}\)

Correct answer:

\(\displaystyle 4\mathbf{i}-2\mathbf{j}+\mathbf{k}\)

Explanation:

Two vectors are defined as orthogonal when their dot product is zero.

 

The dot product of two vectors

 \(\displaystyle \mathbf{u}=u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k}\)   and   \(\displaystyle \mathbf{v}=v_x\mathbf{i}+v_y\mathbf{j}+v_z\mathbf{k}\)

Is given by the expression:

\(\displaystyle \mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y+u_zv_z\)

 

The only vector that satisfies the requirement that the dot product of it and is zero is \(\displaystyle 4\mathbf{i}-2\mathbf{j}+\mathbf{k}\):

\(\displaystyle (3\mathbf{i}+7\mathbf{j}+2\mathbf{k})\cdot (4\mathbf{i}-2\mathbf{j}+\mathbf{k}) =(3)(4)+(7)(-2)+(2)(2)=12-14+2=0\)

Example Question #136 : Dot Product

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

Possible Answers:

\(\displaystyle x^3+y+z\)

\(\displaystyle 3x^2+y^3+z^4\)

\(\displaystyle 3x^3+y^4+5z^5\)

\(\displaystyle 3x+y+5z+x^2+y^3+z^4\)

Correct answer:

\(\displaystyle 3x^3+y^4+5z^5\)

Explanation:

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

\(\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)\). Using the vectors from the problem statement, we get

\(\displaystyle (3x*x^2)+(y*y^3)+(5z*z^4)=3x^3+y^4+5z^5\)

Example Question #137 : Dot Product

Find the dot product between the vectors \(\displaystyle \left \langle -4,-10,1\right \rangle\) and \(\displaystyle \left \langle x,y^2,z\right \rangle\)

Possible Answers:

\(\displaystyle 4-10y+z\)

\(\displaystyle 4x^3-5y^2+z\)

\(\displaystyle 4x^2-10y^2+1\)

\(\displaystyle -4x-10y^2+z\)

Correct answer:

\(\displaystyle -4x-10y^2+z\)

Explanation:

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

\(\displaystyle a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)\). Using the vectors from the problem statement, we get

\(\displaystyle (-4*x)+(-10*y^2)+(1*z)=-4x-10y^2+z\)

Example Question #131 : Dot Product

Solve:

\(\displaystyle \left \langle z, y^3z^2, x\right \rangle \cdot \left \langle 2, e^x, x\right \rangle\)

Possible Answers:

\(\displaystyle 2z+e^xy^3z^2+x^2\)

\(\displaystyle 2z+e^xy^3z^2+x\)

\(\displaystyle 2z+e^xy^3z^2+2x\)

\(\displaystyle \left \langle 2z, e^xy^3z^2, x^2 \right \rangle\)

Correct answer:

\(\displaystyle 2z+e^xy^3z^2+x^2\)

Explanation:

The dot product of two vectors is given by the sum of the products of the corresponding components (for example, \(\displaystyle \left \langle a, b\right \rangle \cdot \left \langle c, d\right \rangle=ac+bd\))

Our final answer is

\(\displaystyle 2z+e^xy^3z^2+x^2\)

Example Question #139 : Dot Product

Find the dot product between the vectors \(\displaystyle \left \langle x,y,4z\right \rangle\) and \(\displaystyle \left \langle 3,-7,1\right \rangle\)

Possible Answers:

\(\displaystyle 3x-7y-z\)

\(\displaystyle 2x-7y+4z\)

\(\displaystyle 3x-7y+4z\)

\(\displaystyle 3x-y+4z\)

Correct answer:

\(\displaystyle 3x-7y+4z\)

Explanation:

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

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

Using the vectors from the problem statement, we get

\(\displaystyle (x*3)+(y*-7)+(4z*1)=3x-7y+4z\)

Example Question #140 : Dot Product

Find the dot product between the vectors \(\displaystyle \left \langle 4,1,-5\right \rangle\) and \(\displaystyle \left \langle 2,-8,1\right \rangle\)

Possible Answers:

\(\displaystyle -5\)

\(\displaystyle 17\)

\(\displaystyle -15\)

\(\displaystyle 5\)

Correct answer:

\(\displaystyle -5\)

Explanation:

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

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

Using the vectors from the problem statement, we get

\(\displaystyle (4*2)+(1*-8)+(-5*1)=-5\)

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