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Question

vertical line a with not stretchy bar on top vertical line equals vertical line b with not stretchy bar on top vertical line equals vertical line c with not stretchy bar on top vertical line equals vertical line a with not stretchy bar on top plus b with not stretchy bar on top plus c with not stretchy bar on top vertical line equals 1  and a with not stretchy bar on top perpendicular b with not stretchy bar on top left parenthesis c with not stretchy bar on top comma a with not stretchy bar on top right parenthesis equals alpha comma left parenthesis c with not stretchy bar on top comma b with not stretchy bar on top right parenthesis equals beta comma t h e n space cos alpha plus cos space beta equals

  1. 3 divided by 2
  2. 1
  3. -1
  4. 2

hintHint:

We are given three vectors having magnitude 1. We are also given that two of the vectors are perpendicular. We are given the the angle between other two vector is α and the angle between the other two is β. We have to find the value of cosα + cosβ. We will use the given conditions to solve the question.

The correct answer is: -1


    The given vectors are a with rightwards arrow on top comma b with rightwards arrow on top a n d c with rightwards arrow on top
    The given conditions are as follows:
    open vertical bar a with rightwards arrow on top close vertical bar equals open vertical bar b with rightwards arrow on top close vertical bar equals open vertical bar c with rightwards arrow on top close vertical bar equals 1
    open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top plus c with rightwards arrow on top close vertical bar equals 1
    a with rightwards arrow on top perpendicular b with rightwards arrow on top
    The angle between the vectors c with rightwards arrow on top space a n d space a with rightwards arrow on top is α.
    The angle between the vectors c with rightwards arrow on top a n d b with rightwards arrow on top is β.
    We have to find the value of cosα + cosβ.
    Let's take dot product of pair of vectors.
    a with rightwards arrow on top. b with rightwards arrow on top equals vertical line a with rightwards arrow on top vertical line open vertical bar b with rightwards arrow on top close vertical bar cos 90 to the power of 0 space
a with rightwards harpoon with barb upwards on top. b with rightwards arrow on top space equals space 0
    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 beta
space space space space space space space space equals left parenthesis 1 right parenthesis left parenthesis 1 right parenthesis cos beta
space space space space space space space space equals cos beta
    c with rightwards arrow on top. a with rightwards arrow on top equals open vertical bar c with rightwards arrow on top close vertical bar open vertical bar a with rightwards arrow on top close vertical bar cos alpha
space space space space space space space equals left parenthesis 1 right parenthesis left parenthesis 1 right parenthesis cos alpha
space space space space space space space equals cos alpha
    Now, we will use the given condition to solve further.
    open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top plus c with rightwards arrow on top close vertical bar equals square root of left parenthesis a with rightwards arrow on top plus b with rightwards arrow on top plus c with rightwards arrow on top right parenthesis. left parenthesis a with rightwards arrow on top plus b with rightwards arrow on top plus c with rightwards arrow on top right parenthesis end root
space space space space space space space space space space space space space space 1 space equals square root of left parenthesis open vertical bar a with rightwards arrow on top close vertical bar squared plus space open vertical bar b with rightwards arrow on top close vertical bar squared plus open vertical bar c with rightwards arrow on top close vertical bar squared plus space 2 left parenthesis a with rightwards arrow on top. b with rightwards arrow on top plus b with rightwards arrow on top. c with rightwards arrow on top plus c with rightwards arrow on top. a with rightwards arrow on top right parenthesis end root
    Substituting all the values we get,
    1 equals square root of 1 plus 1 space plus space 1 space plus 2 cos alpha plus 2 cos beta end root
1 space equals square root of 3 space plus space 2 cos alpha space plus space 2 cos beta end root
S q u a r i n g space b o t h space t h e space s i d e s
1 space equals space 3 space plus space 2 cos alpha space plus space 2 cos beta
minus 2 space equals space 2 cos alpha space plus space 2 cos beta
R e a r r a n g i n g
2 cos alpha space plus space 2 cos beta space equals space minus 2
cos alpha space plus space cos beta space equals space minus 1
    So, the answer is -1.

    For such questions, we should know about the dot product.

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