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Math Help - Vectors

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    Vectors

    Q1: If a = i + 2j - k and b = j + k, find a unit vector perpendicular to both a and b.

    ===

    Q2: The points P and Q have position vectors a + b and 3a - 2b respectively when referred from the origin O. Given that OPQR is a parallelogram, express the vectors PQ and PR in terms of a and b. [I have found PQ = 2a - 3b; PR = a - 4b] By evaluating 2 scalar products, show that if OPQR is a square, then|a| ^2 = 2|b| ^2 .

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    Quote Originally Posted by Tangera View Post
    Q1: If a = i + 2j - k and b = j + k, find a unit vector perpendicular to both a and b.

    ===

    Q2: The points P and Q have position vectors a + b and 3a - 2b respectively when referred from the origin O. Given that OPQR is a parallelogram, express the vectors PQ and PR in terms of a and b. [I have found PQ = 2a - 3b; PR = a - 4b] By evaluating 2 scalar products, show that if OPQR is a square, then|a| ^2 = 2|b| ^2 .

    Thank you for helping!
    1. Take the cross product of \mathbf{a} and \mathbf{b} and then divide this vector by its length.
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    Quote Originally Posted by Tangera View Post
    Q1: If a = i + 2j - k and b = j + k, find a unit vector perpendicular to both a and b.

    ===

    Q2: The points P and Q have position vectors a + b and 3a - 2b respectively when referred from the origin O. Given that OPQR is a parallelogram, express the vectors PQ and PR in terms of a and b. [I have found PQ = 2a - 3b; PR = a - 4b] By evaluating 2 scalar products, show that if OPQR is a square, then|a| ^2 = 2|b| ^2 .

    Thank you for helping!
    2. Drawing a picture always helps.

    Since it's a parallelogram, notice that OR is parallel and of equal magnitude to PQ and that QR is parallel and of equal magnitude to OP.

    What do you have when vectors are parallel and of equal length? They are EQUAL.
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    ^ Thank you for your suggestions! I got stuck at part that requires the evaluation of the scalar product... Do I evaluate PR.OQ = OP.PQ = 0?
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    Quote Originally Posted by Tangera View Post
    ^ Thank you for your suggestions! I got stuck at part that requires the evaluation of the scalar product... Do I evaluate PR.OQ = OP.PQ = 0?
    What sides touch?

    OP touches PQ

    PQ touches QR

    QR touches OR

    OP touches OR.


    If it's a square, then the angles should be right angles. Evaluating any of the dot products of touching sides should give 0 if this is the case.

    Then show that the lengths are equal, and you've got a square.

    See how you go from there.
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    ^ Um...sorry I am still confused...I understand your method, but the question wanted me to show |a|= 2|b| if OPQR is a square...so do I have to equate 2 scalar products?

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    Quote Originally Posted by Tangera View Post
    ^ Um...sorry I am still confused...I understand your method, but the question wanted me to show |a|= 2|b| if OPQR is a square...so do I have to equate 2 scalar products?


    Is that |\mathbf{a}|^2 = 2|\mathbf{b}|^2?
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    ^ Yup the question wanted ...
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