scalar and vector products?

Question

how do i show that any two vectors a and b,

(a · b)² + lla x bll² = llall² llbll

by using scalar and vector products

Answer 1

Just use definitions of scalar and vector products:
ab = |a||b| cos(fi), where fi is the angle between vector a and vector b,
|a x b| = |a||b| sin(fi), |x| is the magnitude of x.
Since cos(fi)^2 + sin(fi)^2 = 1 it follows:
(ab)^2 + |a x b|^2 = |a|^2 * |b|^2.

Answer 2

Understand Scalar Product and Vector Product of Two Vectors with the help of VIDEO explanation. Click on the link to Watch the VIDEO:

http://bit.ly/1yKLqhI

Scalar Product and vector Product of Two Vectors
Dot product or scalar product of two vectors
The Dot product of two vector A and B is defined as the product of the magnitudes of the two vectors multiply by the cosin of the angle between them the scalar product of the two vector is scalar.
If the vectors A and B have the same direction theta is equal to 0 then the scalar product will be equal to the product of the modulii of the vectors as cos 0 is equal to 1. When the vectors are the perpendicular to each other the scalar product is equal to 0 because cos 90 is equal to 0.
Cross product or vector product of two vectors
Let A and B be two vectors and theta be the angle between them. The magnitude of the cross product of the vectors A and B is equal to modulus of vector a into modulus of vector b into sin theta.
The cross product of vector a and b gives the third vector c. The direction of the vector c is normal to the plane containing vectors a and b in such a way that vectors a b and c forms a right handed quardinate system. Cross product can also be written as shown the direction of the vector c can be determined either by using Right Hand Screw Rule or Right Hand Thumb Rule.
Right Hand Screw Rule the screw states that if a right handed screw is rotated from vector A to vector B through a small angle, the direction of the advancing screw gives the direction of the cross product of vectors A and B.
If the Screw is rotated from vector B to Vector A through a small angle the advancement will be downward. Hence cross product of vectors A and B is equal to the negative of the cross product of vectors B and A.
Right Hand Thumb Rule the thumb rule states if the fingers of the right hand curled in such a way that they point along the direction of rotation from vector A to vector B through a small angle, then the thumb points in the direction of the cross product of vectors A and B.
Note that the cross product of the two vectors will not obey commutative law.

Answer 3

I assume you mean:
(a • b)² + lla x bll² = llall² llbll²

Let
θ = angle between the two vectors

Let's start with the left hand side.

Left Hand Side = (a • b)² + lla x bll²

= (||a|| ||b|| cosθ)² + (||a|| ||b|| sinθ)²

= ||a||² ||b||² cos²θ + ||a||² ||b||² sin²θ

= (||a||² ||b||²) (cos²θ + sin²θ)

= ||a||² ||b||² = Right Hand Side

Answer 4

a.b = ||a|| ||b|| cos(theta), and ||a x b|| = ||a|| ||b|| sin(theta) where theta is the angle between a and b. Also sin^2(theta) + cos^2(theta) = 1. That should do it.