What Can Be Said About The Vectors U And V When The Projection Of U Onto V Equals U – The cross product, also called the vector product of two vectors, is written (vec times vec) and is another way to multiply two vectors together.
We get a new vector. This is in contrast to the dot product (or dot product) of two vectors, where the result is a scalar (a number, not a vector!).
What Can Be Said About The Vectors U And V When The Projection Of U Onto V Equals U
In fact, the cross product of two vectors (vec) and (vec) is a “new” vector perpendicular to both (vec) and (vec), we say ( vec times vec ) is normal for the plane containing (vec) and (vec).
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To calculate the vector product or the cross product of two vectors, we use one of these two options:
Given two vectors ( vec = beginu_1 \ u_2 \ u_3 end ) and ( vec = beginv_1 \ v_2 \ v_3 end ),
, (vectimes vec) can be calculated using this formula: [vec times vec = beginu_2v_3 – v_2u_3end vec – beginu_1v_3 -v_1u_3end vec + beginu_1v_2 – v_1 end vec] To avoid subtraction between the first quarter and the second quarter, some mathematics courses (such as
) rewrite this formula as: [vec times vec = beginu_2v_3 – v_2u_3end vec + begin v_1u_3 – u_1v_3end vec + beginu_1v_2 – v_1u_2end vec] Both formulas are
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, (vec times vec) by (vec = start 1 \ -3 \ 2 end) and (vec = start 4 \ 0 \ 6 end) .
Using the second formula, given above: [vec times vec = beginu_2v_3 – v_2u_3end vec + begin v_1u_3 – u_1v_3end vec + beginu_1v_2 – v_1u_2end vec] (u_1) With = 1 ), (u_2 = -3 ), (u_3 = 2 ), (v_1 = 4 ), (v_2 = 0 ) and (v_3 = 6 ), we find: [begin vectimes vec & = begin-3 times 6 – 0 times 2 endvec + begin4 times 2 – 1 times 6 endvec + begin1 times 0 – 4 times (-3)end vec \ & = begin-18 -0endvec + begin8 – 6 end vec + begin0+12endvec \ vectimes vec & = -18 vec + 2 vec + 12 vec end ]
, (vectimes vec) is equal to: [ vectimes vec = begin vec & vec & vec \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 end ] Note that:
(vec = start 2 \ -1 \ 3 end) and (vec = start 5 \ 0 \ 4 end), we learn how to calculate the cross product, ( vec times vec), using matrix algebra.
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Given ( vec = start 2 \ 1 \ -3 end ) and ( vec = start 4 \ 0 \ 5 end ), calculate their cross product (vec times vec).
Using the method we saw, we can set: [begin vec times vec & = begin vec & vec & vec \ 2 & 1 & -3 \ 4 & 0 & 5 end & = vec begin 1 & -3 \ 0 & 5 end – vec begin 2 & -3 \ 4 & 5end + vec begin 2 & 1 \ 4 & 0 end & = vec start 1 times 5 – 0 times (-3)end – vec start 2times 5 – 4 times (-3) end + vec start 2 times 0 – 4 times 1 \ end \ & = vecstart 5 – 0end – vec start 10 – (-12) end + vec start 0 – 4 end \ & = vec start 5 end – vec start 10 + 12 end + vec start – 4 end \ vec times vec & = 5 vec – 22 vec – 4 vec end ] It is our final answer, we can now say that (vec times vec = 5 vec – 22 vec – 4 vec ).
, two vectors can be used to calculate the area of a parallelogram and the area of a triangle.
Given a parallelogram, whose sides are defined by two vectors (vec) and (vec), the area is given by: [text = begin vectimes vecend ] means :
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A parallelogram has side lengths defined by vectors (vec = \ -1 \ 3 end) and (vec = start 1 \ 5 \ 4 end).
Given a triangle, whose sides are defined by two vectors (vec) and (vec), the area is given by: [text = frac begin vectimes vec end] Note: This is half the area of a parallelogram whose sides are (vec) and (vec).
A triangle has two sides defined by the vector (vec = start -1 \ 7 \ 1 end ) and ( vec = start 1 \ 4 \ 3 end ).
