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In this lesson, we will learn how to find the angle between two straight lines in three dimensions.

Q1:

A straight line πΏ 1 passes through the two points π΄ ( β 2 , 2 , β 3 ) and π΅ ( β 6 , β 4 , β 5 ) , and a straight line πΏ 2 passes through the two points πΆ ( 1 , 4 , 1 ) and π· ( β 9 , β 6 , β 9 ) . Find the measure of the angle between the two lines, giving your answer to two decimal places if necessary.

Q2:

Find, to the nearest second, the measure of the angle between the straight line π₯ + 1 2 = π¦ β 2 β 4 = π§ + 2 5 and the positive direction of the π₯ -axis.

Q3:

Determine, to the nearest second, the measure of the angle between the two lines that have direction ratios of ( β 4 , β 3 , β 4 ) and ( β 3 , β 3 , β 1 ) .

Q4:

Find the measure of the angle between the straight line π₯ = 1 , π¦ = 2 and the straight line π¦ = β 1 , π§ = 0 .

Q5:

Find, to the nearest second, the measure of the angle between the two straight lines β 2 π₯ = 4 π¦ = β 3 π§ and β 4 π₯ = β 5 π¦ = 2 π§ .

Q6:

Find the measure of the angle between the two straight lines πΏ βΆ π₯ = 5 β 8 π‘ ο§ , π¦ = β 3 β 4 π‘ , π§ = 5 + 6 π‘ and πΏ βΆ π₯ β 5 3 = π¦ + 5 β 6 = π§ β 2 β 2 ο¨ , and round it to the nearest second.

Q7:

Find the measure of the angle between the two straight lines whose direction cosines are οΏ 3 1 3 β 2 , 9 1 1 β 2 , β 3 7 β 2 ο and οΏ β 1 0 1 3 β 2 , β 8 1 3 β 2 , 9 8 β 2 ο . Give your answer to the nearest second.

Q8:

Find the measure of the angle between the two straight lines β π = οΌ 2 7 , β 2 3 , β 1 ο + π‘ οΌ β 2 7 , β 4 3 , 9 5 ο 1 1 and β 6 π₯ β 2 7 = 4 π¦ β 3 β 6 = 3 β 8 π§ β 5 .

Q9:

If πΏ βΆ π₯ = 0 1 , π¦ = π§ and πΏ βΆ π¦ = 0 2 , π₯ = π§ , then find the value of π .

Q10:

Given that a straight line passes through the origin and the point ( 4 , 1 , 2 ) , determine the exact value of c o s π π§ .

Note that π π§ is the measure of the angle between the straight line and the positive direction of the π§ -axis.

Q11:

If a straight line makes direction angles of measures 6 0 β with the π¦ -axis and 6 0 β with the π§ -axis, then find the direction angle it makes with the π₯ -axis.

Q12:

Find, to the nearest second, the measure of the angle between the straight line π₯ + 1 2 = π¦ β 1 β 3 = π§ + 4 4 and the positive direction of the π₯ -axis.

Q13:

Find, to the nearest second, the measure of the angle between the straight line π₯ + 1 β 2 = π¦ β β 2 β 3 = π§ + 1 1 and the positive direction of the π§ -axis.

Q14:

Find, to the nearest second, the measure of the angle between the two straight lines 3 π₯ = β 2 π¦ = β π§ and 5 π₯ = π¦ = 5 π§ .

Q15:

Find, to the nearest second, the measure of the angle between the two straight lines β 5 π₯ = π¦ = 3 π§ and β 3 π₯ = β 3 π¦ = 4 π§ .

Q16:

A straight line πΏ 1 passes through the two points π΄ ( β 6 , 5 , β 4 ) and π΅ ( β 8 , β 7 , β 8 ) , and a straight line πΏ 2 passes through the two points πΆ ( β 6 , 1 , β 5 ) and π· ( β 8 , β 3 , β 7 ) . Find the measure of the angle between the two lines, giving your answer to two decimal places if necessary.

Q17:

A straight line πΏ 1 passes through the two points π΄ ( 5 , 1 , 9 ) and π΅ ( β 7 , β 3 , β 1 ) , and a straight line πΏ 2 passes through the two points πΆ ( 1 , 7 , 6 ) and π· ( β 3 , β 9 , 2 ) . Find the measure of the angle between the two lines, giving your answer to two decimal places if necessary.

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