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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