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In this lesson, we will learn how to find the angle between two vectors in space using their dot product.

Q1:

If , , and , where is a unit vector, find the measure of the angle between and rounded to the nearest minute given .

Q2:

Find the angle π between the vectors V = β¨ 5 , 1 , β 2 β© and W = β¨ 4 , β 4 , 3 β© . Give your answer correct to two decimal places.

Q3:

The angle between A and B is 2 2 β . If | | = 3 | | = 2 5 . 2 A B , find A B β to the nearest hundredth.

Q4:

Find .

Q5:

Find the angle between the vectors and . Give your answer correct to two decimal places.

Q6:

Find the angle π between the vectors V = β¨ 2 , 1 , 4 β© and W = β¨ 1 , β 2 , 0 β© .

Q7:

Given that , , and , determine the size of the smaller angle between the two vectors.

Q8:

Given that and , determine, to the nearest hundredth, the size of the smaller angle between the two vectors.

Q9:

Q10:

Given , , , and , determine the size of the angle between vectors and approximated to the nearest hundredth.

Q11:

Q12:

Q13:

If and , find the size of the angle between the two vectors approximated to the nearest hundredth.

Q14:

Q15:

Find the angle π between the vectors V = β¨ 4 , 2 , β 1 β© and W = β¨ 8 , 4 , β 2 β© .

Q16:

Find the value of π₯ given A = ο± 4 π , π₯ , π ο΅ c o s l o g s i n 3 , B = ο± π , 1 6 , 4 π ο΅ c o s l o g s i n 2 and A B β = 1 0 where π is the angle between the vectors A and B . Give the answer to two decimal places.

Q17:

Find the angle π between the vectors V i j k = β + 2 + and W i j k = β 3 + 6 + 3 .

Q18:

Find the angle π between the vectors V = β¨ 7 , 2 , β 1 0 β© and W = β¨ 2 , 6 , 4 β© . Give your answer correct to one decimal place.

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