Suppose vectors a&b if a x b=c. Show that c is perpendicular to a&b. Please help me.
it is more definitive than this. c is perpendicular to the plane a and b lie in.
you have to start with the definition of cross product
i x j=k
j x k=i
k x i=j
if a= a1 i + a2 j +a3 k and
if b= b1 i + b2 j +b3 k
then you can demonstrate with algebra (do it longhand) that
a x b= the matrix expanded for
i;j;k
a1;a2;a3
b1;b2;b3
if you expand that as a determinant,you get axb
Now notice each component of that determinant (I will do i)
(a2b3-b2a3)i Notice this is i component is the product of the j,k components of the a,b vectors. This component of the cross product is perpendicular to the cross product result.
Thank u so much for your help!
To show that the vector c is perpendicular to vectors a and b, we need to show that the dot product of c with each of a and b is zero. Here is how you can demonstrate this:
Given:
- Vectors a and b: a = (a1, a2, a3) and b = (b1, b2, b3)
- Vector c, the result of the cross product of vectors a and b: c = (c1, c2, c3)
The dot product of c with a is given by:
c · a = (c1, c2, c3) · (a1, a2, a3) = c1 * a1 + c2 * a2 + c3 * a3
Similarly, the dot product of c with b is given by:
c · b = (c1, c2, c3) · (b1, b2, b3) = c1 * b1 + c2 * b2 + c3 * b3
To show that c is perpendicular to a and b, we need to demonstrate that c · a = 0 and c · b = 0. If both of these dot products equal zero, it indicates that c is orthogonal (perpendicular) to both a and b.
Using the properties of the cross product, we can determine the components c1, c2, and c3 of vector c. The cross product can be computed as follows:
c1 = a2 * b3 - a3 * b2
c2 = a3 * b1 - a1 * b3
c3 = a1 * b2 - a2 * b1
Calculate these values for c1, c2, and c3 using the given vectors a and b. Then substitute these values into the dot product equations for c · a and c · b and simplify the expressions.
If both dot products are indeed zero, it can be concluded that the vector c is perpendicular to both a and b.