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Question: A line AB in three dimensional space making an angle \({45^\circ }\) and \({120^\circ }\) with the p...

A line AB in three dimensional space making an angle 45∘{45^\circ } and 120∘{120^\circ } with the positive x-axis and the positive y-axis respectively . If AB makes an acute angle e with the positive z-axis , then e equals to :
A) 45∘{45^\circ }
B) 60∘{60^\circ }
C) 75∘{75^\circ }
D) 30∘{30^\circ }

Explanation

Solution

As we know that if the line AB makes an angle (a , b ,c ) with the positive (x , y , z) axis then cos⁡2a+cos⁡2b+cos⁡2c=1{\cos ^2}a + {\cos ^2}b + {\cos ^2}c = 1 hence in this question two angles are know third one is find out from this formula .

Complete step-by-step answer:
In the question it is given that Line AB is in three dimensional making angles 45∘{45^\circ } and 120∘{120^\circ } with the positive x-axis and the positive y-axis respectively .
And makes an acute angle e with the positive z-axis mean that angle e < 90∘{90^\circ } .
Hence from the Vector property we know that if the line AB makes an angle (a , b ,c ) with the positive (x , y , z) axis then cos⁡2a+cos⁡2b+cos⁡2c=1{\cos ^2}a + {\cos ^2}b + {\cos ^2}c = 1 .
Hence from the question a = 45∘{45^\circ } , b = 120∘{120^\circ }and c = e ,
cos⁡245∘+cos⁡2120∘+cos⁡2e=1{\cos ^2}{45^\circ } + {\cos ^2}{120^\circ } + {\cos ^2}e = 1
cos⁡2e=1−(cos⁡245∘+cos⁡2120∘){\cos ^2}e = 1 - ({\cos ^2}{45^\circ } + {\cos ^2}{120^\circ })
cos⁡2e=1−(12)2−(−12)2{\cos ^2}e = 1 - {\left( {\dfrac{1}{{\sqrt 2 }}} \right)^2} - {\left( {\dfrac{{ - 1}}{2}} \right)^2}
cos⁡2e=1−(12)−(14){\cos ^2}e = 1 - \left( {\dfrac{1}{2}} \right) - \left( {\dfrac{1}{4}} \right)
cos⁡2e=1−(34){\cos ^2}e = 1 - \left( {\dfrac{3}{4}} \right)
cos⁡2e=(14){\cos ^2}e = \left( {\dfrac{1}{4}} \right)
cos⁡e=±(12)\cos e = \pm \left( {\dfrac{1}{2}} \right)
For the acute angle e < 90∘{90^\circ } hence only positive will be considered .
cos⁡e=(12)\cos e = \left( {\dfrac{1}{2}} \right)
e = 60∘{60^\circ }

Hence option B will be the correct answer.

Note: In this question it is given in that the angle is e is acute angle , If it is obtuse angle then we have to consider the negative value of cos⁡e\cos ethat is cos⁡e=−(12)\cos e = - \left( {\dfrac{1}{2}} \right) Hence the e=120∘e = {120^\circ } .
Multiplication of a vector by a scalar quantity is called Scaling. In this type of multiplication, only the magnitude of a vector is changed not the direction.
Always remember that if a line makes an angle (a , b ,c ) with the positive (x , y , z) axis then cos⁡2a+cos⁡2b+cos⁡2c=1{\cos ^2}a + {\cos ^2}b + {\cos ^2}c = 1 .