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Math Help - Proving Vectors are collinear under certain conditions

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    Proving Vectors are collinear under certain conditions

    Three points X, Y, Z have position vectors x,y,z. Show that X, Y and Z are collinear iff x ^ y + y ^ z + z ^ x = 0.

    Here "^" denotes the vector cross product
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    Quote Originally Posted by Paul616 View Post
    Three points X, Y, Z have position vectors x,y,z. Show that X, Y and Z are collinear iff x ^ y + y ^ z + z ^ x = 0.

    Here "^" denotes the vector cross product
    If X, Y and Z are collinear then z=\lambda x + (1-\lambda)y for some scalar \lambda. You can then verify that y\times z + z\times x + x\times y=0 (remembering that the cross product of a vector with itself is always 0).

    For the converse, if y\times z + z\times x + x\times y=0 then 0 = x.(y\times z) + x.(z\times x) + x.(x\times y) = x.(y\times z) (since the other two terms are 0). So y\times z is orthogonal to x. But it is also orthogonal to y and z. If x, y and z are linearly independent then y\times z is orthogonal to the whole space and is therefore 0. But that would mean that y and z are not linearly independent. That contradiction shows that the three vectors must be linearly dependent.

    So one of them, z say, is a linear combination of the others, z = \lambda x + \mu y. Substitute that value for z into the equation y\times z + z\times x + x\times y=0 and you will find that \mu = 1-\lambda, which is the condition for X, Y and Z to be collinear.
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