Maths-
General
Easy

Question

Let the unit vectors a with not stretchy rightwards arrow on top and b with not stretchy rightwards arrow on top be perpendicular and the unit vector straight C with rightwards arrow on top be inclined at an angle θ to both a with not stretchy rightwards arrow on top and straight b with rightwards arrow on top Ifstraight c with rightwards arrow on top equals alpha straight a with rightwards arrow on top plus beta straight b with rightwards arrow on top plus gamma left parenthesis straight a with rightwards arrow on top cross times straight b with rightwards arrow on top right parenthesis  then

  1. alpha equals beta equals cos invisible function application theta comma gamma squared equals cos invisible function application 2 theta
  2. alpha equals beta equals cos invisible function application theta comma gamma squared equals negative cos invisible function application 2 theta
  3. alpha equals cos invisible function application theta comma beta equals sin invisible function application theta comma gamma squared equals cos invisible function application 2 theta
  4. None of these

hintHint:

We are given three unit vectors. Two of the vectors are perpendicular and the two vectors make same angle with the third be vector. We are given the third vector. We have to find the values of the variables of the third vector.

The correct answer is: alpha equals beta equals cos invisible function application theta comma gamma squared equals negative cos invisible function application 2 theta


    The given unit vectors are a with rightwards arrow on top comma b with rightwards arrow on top a n d c with rightwards arrow on top
    So, open vertical bar a with rightwards arrow on top open vertical bar equals close vertical bar b with rightwards arrow on top open vertical bar equals close vertical bar c with rightwards arrow on top close vertical bar space equals 1
    We are given that a with rightwards arrow on top perpendicular b with rightwards arrow on top
    The vectors a with rightwards arrow on top a n d b with rightwards arrow on top makes angle θ with the vector c with rightwards arrow on top.
    The vector c with rightwards arrow on top is given as
    c with rightwards arrow on top equals alpha a with rightwards arrow on top plus beta b with rightwards arrow on top plus gamma open parentheses a with rightwards arrow on top cross times b with rightwards arrow on top close parentheses
    We have to find the values of alpha comma beta space a n d space gamma.
    Let's take the dot product of the vectors.
    a with rightwards arrow on top. b with rightwards arrow on top equals 0      ...( Both are perpendicular)
    b with rightwards arrow on top. c with rightwards arrow on top equals open vertical bar b with rightwards arrow on top close vertical bar open vertical bar c with rightwards arrow on top close vertical bar cos theta
space space space space space space space space equals left parenthesis 1 right parenthesis left parenthesis 1 right parenthesis cos theta space space
space space space space space space space space equals cos theta
    Let's substitute value of c with rightwards arrow on top
    a with rightwards arrow on top. open square brackets alpha a with rightwards arrow on top plus beta b with rightwards arrow on top plus gamma left parenthesis a with rightwards arrow on top cross times b with rightwards arrow on top right parenthesis close square brackets space equals space cos theta space

N o w comma space a with rightwards arrow on top. b with rightwards arrow on top equals 0
A n d comma space t h e space v e c t o r space left parenthesis a with rightwards arrow on top cross times b with rightwards arrow on top right parenthesis space w i l l space b e space p e r p e d i c u l a r space t o space a with rightwards arrow on top. space
S o comma space t h e i r space d o t space p r o d u c t space w i l l space b e space z e r o.

alpha left parenthesis a with rightwards arrow on top. a with rightwards arrow on top right parenthesis space plus space 0 space plus space 0 space equals space cos theta
alpha left parenthesis 1 right parenthesis space equals space cos theta
alpha space equals cos theta
    a with rightwards arrow on top. c with rightwards arrow on top equals open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar cos theta
space space space space space space space space equals left parenthesis 1 right parenthesis left parenthesis 1 right parenthesis cos theta
space space space space space space space space equals cos theta
    We will substitute the value of c with rightwards arrow on top
    b with rightwards arrow on top. left square bracket alpha a with rightwards arrow on top plus beta b with rightwards arrow on top plus gamma left parenthesis a with rightwards arrow on top cross times b with rightwards arrow on top right parenthesis space equals space cos theta
space
b with rightwards arrow on top. a with rightwards arrow on top space equals space 0
left parenthesis a with rightwards arrow on top cross times b with rightwards arrow on top right parenthesis space w i l l space b e space p e r p e d i c u l a r space t o space b with rightwards arrow on top
space space
S o comma space beta left parenthesis b with rightwards arrow on top. b with rightwards arrow on top right parenthesis space equals space cos theta
space space space space space space space beta left parenthesis 1 right parenthesis space equals space cos theta
space space space space space space space space space space beta space equals space cos theta
    Now, we will do following multiplication 
c with rightwards arrow on top. c with rightwards arrow on top equals left square bracket stack alpha a with rightwards arrow on top plus beta b with rightwards arrow on top plus gamma left parenthesis a with rightwards arrow on top cross times b with rightwards arrow on top right parenthesis right square bracket. left square bracket alpha a with rightwards arrow on top space plus space beta b with rightwards arrow on top space plus gamma left parenthesis a with rightwards arrow on top space cross times space b with rightwards arrow on top right parenthesis right square bracket
space space space space space 1 space equals space alpha squared space plus space beta squared space plus space gamma squared
space space space space space 1 space equals space cos squared theta space plus space cos squared theta space plus space gamma squared
space space space space space 1 space equals 2 cos squared theta space plus space gamma squared
space space space space space space 1 space minus space 2 cos squared theta space equals gamma squared
space space space space space space space space space space space cos 2 theta space equals gamma squared
space space space space space space space space space space space space space space space gamma squared space equals space cos 2 theta
    We have all the three values now.

    We should know how to take dot product of two vectors.

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