Question Video: Finding the Direction Angles of a Given Vector | Nagwa Question Video: Finding the Direction Angles of a Given Vector | Nagwa

Question Video: Finding the Direction Angles of a Given Vector Mathematics • Third Year of Secondary School

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Find the direction angles of the vector <21/2, (21√2)/2, 21/2>.

03:10

Video Transcript

Find the direction angles of the vector 21 over two, 21 root two over two, 21 over two.

We begin by recalling that if vector 𝐀 has components 𝐴 sub 𝑥, 𝐴 sub 𝑦, and 𝐴 sub 𝑧, then the direction cosines are cos 𝛼 is equal to 𝐴 sub 𝑥 over the magnitude of vector 𝐀, cos 𝛽 is equal to 𝐴 sub 𝑦 over the magnitude of vector 𝐀, and cos 𝛾 is equal to 𝐴 sub 𝑧 over the magnitude of vector 𝐀, where 𝛼, 𝛽, and 𝛾 are the direction angles we’re trying to calculate in this question.

If we let 𝐀 be the vector in this question, we will begin by calculating its magnitude. This is calculated by finding the sum of the squares of the individual components and then square rooting the answer. The magnitude of vector 𝐀 is therefore equal to the square root of 441 over four plus 882 over four plus 441 over four. This simplifies to the square root of 441. And since the magnitude must be positive, this is equal to 21.

We can now find expressions for cos 𝛼, cos 𝛽, and cos 𝛾. cos 𝛼 is equal to 21 over two over 21, which simplifies to one-half. cos 𝛽 is equal to 21 root two over two divided by 21. Once again, we can divide the numerator and denominator by 21 such that cos 𝛽 is equal to root two over two. Since the third component of vector 𝐀 is the same as the first, cos 𝛾 is also equal to one-half.

We are now in a position where we can calculate the angles 𝛼, 𝛽, and 𝛾. We can do this by taking the inverse cosine of each equation or by using our knowledge of special angles. Noting that our angles must lie between zero and 180 degrees, 𝛼 is equal to 60 degrees, 𝛽 is equal to 45 degrees, and 𝛾 is equal to 60 degrees. The direction angles of the vector 21 over two, 21 root two over two, 21 over two are 60 degrees, 45 degrees, and 60 degrees.

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