Suppose vectors a&b, if a cross b=c. Show that c is perpendicular to a&b.
To show that vector c is perpendicular to vectors a and b, we need to demonstrate that the dot product between c and each of a and b is zero.
The dot product of two vectors a and b is defined as follows:
a · b = ax * bx + ay * by + az * bz
where ax, ay, and az are the components of vector a, and bx, by, and bz are the components of vector b.
Now, let's take the cross product of vectors a and b:
a x b = (ay * bz - az * by) * i - (ax * bz - az * bx) * j + (ax * by - ay * bx) * k
The components of vector c are determined by the above expression.
To show that c is perpendicular to a, we need to take the dot product of c and a and demonstrate that it equals zero:
c · a = [(ay * bz - az * by) * i - (ax * bz - az * bx) * j + (ax * by - ay * bx) * k] · (ax * i + ay * j + az * k)
Expanding the dot product:
c · a = (ay * bz - az * by) * (ax) + (ax * bz - az * bx) * (ay) + (ax * by - ay * bx) * (az)
Similarly, we also need to show that c · b equals zero:
c · b = (ay * bz - az * by) * (bx) + (ax * bz - az * bx) * (by) + (ax * by - ay * bx) * (bz)
Thus, to prove that c is perpendicular to both a and b, we have to show that c · a and c · b are both equal to zero, i.e.,:
(ay * bz - az * by) * (ax) + (ax * bz - az * bx) * (ay) + (ax * by - ay * bx) = 0
and
(ay * bz - az * by) * (bx) + (ax * bz - az * bx) * (by) + (ax * by - ay * bx) = 0
By confirming that c · a and c · b are both zero, we can conclude that vector c is perpendicular to both vectors a and b.