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In this lesson, we will learn how to find the direction cosine of a vector or a straight line from the direction ratios or that of a normal to a plane given the plane equation.

Q1:

Which of the following gives the direction cosines of a straight line?

Q2:

Find the direction cosines of the straight line whose direction ratio is 2 ∶ − 1 ∶ 1 .

Q3:

The given figure represents a vector 𝑂 𝐴 whose norm is 5 units. Find the measure of the direction angles of 𝑂 𝐴 rounded to one decimal place.

Q4:

Suppose that ‖ ‖ ⃑ 𝐴 ‖ ‖ = 6 and ⃑ 𝐴 has direction cosines 2 3 , − 2 3 , and − 1 3 . Determine ⃑ 𝐴 × ⃑ 𝐵 , where ⃑ 𝐵 = ( − 8 , 0 , 3 ) .

Q5:

Find the direction cosines of the normal to the plane 4 𝑥 + 8 𝑦 − 3 𝑧 = 2 8 .

Q6:

If the direction cosines of a straight line are 1 𝑐 , 1 𝑐 , 1 𝑐 , then find the possible values of 𝑐 .

Q7:

Given that 𝑙 , 𝑚 , and 𝑛 are the direction cosines of a straight line, find the value of 𝑙 + 𝑚 + 𝑛 2 2 2 .

Q8:

If the direction angles of a straight line are 𝑇 𝑥 , 𝑇 𝑦 , and 𝑇 𝑧 , then find the value of c o s c o s c o s 2 𝑇 + 2 𝑇 + 2 𝑇 𝑥 𝑦 𝑧 .

Q9:

Find the direction cosines of the straight line whose direction ratio is 4 ∶ 5 ∶ − 2 .

Q10:

Find the direction cosines of the normal to the plane 8 𝑥 + 5 𝑦 − 2 𝑧 = 6 .

Q11:

Find the direction cosines of the normal to the plane 6 𝑥 + 7 𝑦 + 3 𝑧 = − 6 .

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