Q: Given vector $\vec{a} = 6\hat{i} + 12\hat{j}$ and vector $\vec{b} = 10\hat{i} + 25\hat{j}$, calculate $\vec{a} \cdot \vec{b}$ using $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$.

$6 \times 10 + 12 \times 25 = 60 + 300 = 360$

Q: Given vector $\vec{a} = 15\hat{i}$ and vector $\vec{b} = 15\hat{j}$, calculate $\vec{a} \cdot \vec{b}$ using $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$.

$15 \times 0 + 0 \times 15 = 0$

Q: Given vector $\vec{a} = 24\hat{i}$ and vector $\vec{b} = 24\hat{j}$, calculate $\vec{a} \cdot \vec{b}$ using $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$.

$24 \times 0 + 0 \times 24 = 0$

Q: If $\vec{a} = 3\hat{i} + 4\hat{j}$ and $\vec{b} = 4\hat{i} + 3\hat{j}$, what is $\vec{a} \cdot \vec{b}$?

$3 \times 4 + 4 \times 3 = 12 + 12 = 24$

Q: A force $\vec{F} = 5\hat{i} + 12\hat{j}$ N acts on a point that moves $\vec{d} = 2\hat{i} + 3\hat{j}$ m. What is the work done by the force?

$5 \times 2 + 12 \times 3 = 10 + 36 = 46$ J

Q: If $\vec{a} = 7\hat{i} - 2\hat{j}$ and $\vec{b} = -3\hat{i} + 5\hat{j}$, what is $\vec{a} \cdot \vec{b}$?

$7 \times -3 + (-2) \times 5 = -21 - 10 = -31$

Q: Which of the following is NOT a property of the dot product?

$\vec{a} \cdot \vec{b} = \vec{a} \times \vec{b}$

Q: If $\vec{a} = 8\hat{i} + 6\hat{j}$ and $\vec{b} = 2\hat{i} - 3\hat{j}$, what is $\vec{a} \cdot \vec{b}$?

$8 \times 2 + 6 \times (-3) = 16 - 18 = -2$

Question 1

Given vector $\vec{a} = 6\hat{i} + 12\hat{j}$ and vector $\vec{b} = 10\hat{i} + 25\hat{j}$, calculate $\vec{a} \cdot \vec{b}$ using $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$.

Accepted Answer

$6 	imes 10 + 12 	imes 25 = 60 + 300 = 360$

Question 2

Given vector $\vec{a} = 15\hat{i}$ and vector $\vec{b} = 15\hat{j}$, calculate $\vec{a} \cdot \vec{b}$ using $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$.

Accepted Answer

$15 	imes 0 + 0 	imes 15 = 0$

Question 3

Given vector $\vec{a} = 24\hat{i}$ and vector $\vec{b} = 24\hat{j}$, calculate $\vec{a} \cdot \vec{b}$ using $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$.

Accepted Answer

$24 	imes 0 + 0 	imes 24 = 0$

Question 4

If $\vec{a} = a_1\hat{i} + a_2\hat{j}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j}$, which formula correctly calculates the dot product $\vec{a} \cdot \vec{b}$?

Accepted Answer

$a_1b_1 + a_2b_2$

Question 5

Which of the following statements about the dot product of two perpendicular vectors is correct?

Accepted Answer

The dot product is zero.

Question 6

If $\vec{a} = 3\hat{i} + 4\hat{j}$ and $\vec{b} = 4\hat{i} + 3\hat{j}$, what is $\vec{a} \cdot \vec{b}$?

Accepted Answer

$3 	imes 4 + 4 	imes 3 = 12 + 12 = 24$

Question 7

A force $\vec{F} = 5\hat{i} + 12\hat{j}$ N acts on a point that moves $\vec{d} = 2\hat{i} + 3\hat{j}$ m. What is the work done by the force?

Accepted Answer

$5 	imes 2 + 12 	imes 3 = 10 + 36 = 46$ J

Question 8

If $\vec{a} = 7\hat{i} - 2\hat{j}$ and $\vec{b} = -3\hat{i} + 5\hat{j}$, what is $\vec{a} \cdot \vec{b}$?

Accepted Answer

$7 	imes -3 + (-2) 	imes 5 = -21 - 10 = -31$

Question 9

Which of the following is NOT a property of the dot product?

Accepted Answer

$\vec{a} \cdot \vec{b} = \vec{a} 	imes \vec{b}$

Question 10

If $\vec{a} = 8\hat{i} + 6\hat{j}$ and $\vec{b} = 2\hat{i} - 3\hat{j}$, what is $\vec{a} \cdot \vec{b}$?

Accepted Answer

$8 	imes 2 + 6 	imes (-3) = 16 - 18 = -2$

Unit Vectors and Magnitude in Trigonometry

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