Given the vectors: A = 3i + 2j – k and B = 5i +5j, find:
- The cross product A×B.
- The area of the parallelogram spanned by A and B.
- The y and z components of a vector C = 2i + Cy j +Cz k parallel to B.
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Solution:
It is essential when working with vectors to use proper notation. Always draw an arrow over the letters representing vectors. You can also use bold characters to represent a vector quantity.
The cross product of two vectors is a vector given by the following determinant:
And substituting the components of A and B:
The cross product is always perpendicular to both vectors. You can verify it by performing the dot product of each vector and the result of their cross product. Both have to be zero.
If we call D = A×B:
And doing the same for vector B:
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The area of the parallelogram spanned by A and B is the magnitude (modulus) of their cross vector:
If vector C is parallel to B their cross product must be zero:
And expanding the previous determinant:
But for a vector to be zero its components must be zero too:
Therefore, C expressed in unit vector notation is given by:
