|axb| = |a| |b| sinθ
a.b = |a| |b| cosθ
now square and add them up
a.b = |a| |b| cosθ
now square and add them up
First, let's start with the left side of the equation:
|a x b|^2 + |a . b|^2
The cross product of two vectors, a and b, can be defined as:
|a x b| = |a| |b| sin(theta)
where |a| and |b| are the magnitudes of vectors a and b, and theta is the angle between the two vectors.
The dot product of two vectors, a and b, can be defined as:
|a . b| = |a| |b| cos(theta)
where |a| and |b| are the magnitudes of vectors a and b, and theta is the angle between the two vectors.
Now, let's substitute these values into the left side of the equation:
|a x b|^2 + |a . b|^2
= (|a| |b| sin(theta))^2 + (|a| |b| cos(theta))^2
Expanding and simplifying this expression, we get:
= |a|^2 |b|^2 sin^2(theta) + |a|^2 |b|^2 cos^2(theta)
Using the trigonometric identity sin^2(theta) + cos^2(theta) = 1, we can rewrite the above expression as:
= |a|^2 |b|^2 (sin^2(theta) + cos^2(theta))
Now, sin^2(theta) + cos^2(theta) equals 1, so we have:
= |a|^2 |b|^2 (1)
Since anything multiplied by 1 is equal to itself, we can simplify the expression further:
= |a|^2 |b|^2
This is the same as the right side of the equation.
Therefore, we have shown that |a x b|^2 + |a . b|^2 = |a|^2 |b|^2.