Question

angle between (P+Q)and (P-Q)will be

Answer 1

Explanation:

Suppose the angle between the vectors p and q is α and the angle between p+q and p-q is θ .

The scalar product of these two vectors p+q and p-q is

(p+q).(p-q) = p^2 - q^2

= |p|^2 -|q|^2 .....(1)

The magnitude of the scalar product can be expressed in another form, i.e.-

|p+q| . |p-q| cosθ

= sqrt(|p|^2+|q|^2+2pq cosα) x

sqrt(|p|^2+|q|^2-2pq cosα) x cosθ

=sqrt[(|p|^2 + |q|^2)^2 -4p^2q^2cos^2α] cosθ .....(2)

From (1) and (2)

|p|^2 -|q|^2 =

sqrt[(|p|^2 + |q|^2)^2 -4p^2q^2cos^2α] cosθ

Therefore

cosθ =

(|p|^2 -|q|^2)

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sqrt[(|p|^2+|q|^2)^2 - 4p^2q^2cos^2α]

When p and q are perpendicular to each other, or α= 90 degrees

Then

cosα=0

And

cosθ =

(|p|^2 -|q|^2)

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(|p|^2+|q|^2)