Maths-
General
Easy

Question

The acute angle made by the line joining the points left parenthesis 1 comma negative 3 , 2 right parenthesis and left parenthesis 3 comma negative 5 , 1 right parenthesis with the Y-axis is

  1. c o s to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 2 over denominator 3 end fraction close parentheses    
  2. c o s to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator 3 end fraction close parentheses    
  3. c o s to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 2 over denominator 5 end fraction close parentheses    
  4. c o s to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 3 over denominator 5 end fraction close parentheses    

Hint:

We are given two points. We have to find the angle made by this line with the y-axis. We will use the direction ratios and direction cosine the find the answer.

The correct answer is: c o s to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 2 over denominator 3 end fraction close parentheses


    The given points are (1, -3, 2) and (3, -5, 1)
    We have to find the angle it makes with y-axis.
    Let the angle made by the line with the x, y and z axis be
    Direction ratios: Let’s represent direction ratios as a, b and c. If the given points are (x1, y1, z1) and (x2, y2, z2) then
    a = x1 – x2
    = 1 – 3
    = -2
    b = y1 – y2
    = -3 –(-5)
    = -3 + 5
    = 2
    c = z1 – z2
    = 2 – 1
    = 1
    Direction cosine of angle made with y axis will be given as

    cos beta equals fraction numerator b over denominator square root of a squared plus b squared plus c squared end root end fraction
space space space space space space space space space equals space fraction numerator 2 over denominator square root of 2 squared plus 2 squared plus 1 squared end root end fraction
space space space space space space space space space equals fraction numerator 2 over denominator square root of 9 end fraction
space space space space space space space space space equals 2 over 3
space
beta space equals space cos to the power of negative 1 end exponent left parenthesis 2 over 3 right parenthesis
    This is the required answer.

    We should remember the formula of direction cosines of all the angles.

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space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 3 fraction numerator x cubed plus y cubed plus z cubed minus 3 x y z over denominator x cubed plus y cubed plus y cubed minus 3 x y z end fraction
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