The sum & difference of two non-zero vectors Ä & B are equal in magnitude.what can you conclude about these vectors?why?
Sum A+B
difference=A-B
maginitude sum=ABcosTheta
magnitude difference=ABcos(PI-Theta)
setting them equal, then cosTheta=cos(PI-Theta)
costheta=sinThetasinPI-cosThetaCosPI
cosTheta=sinTheta+cosTheta
which makes sense only if theta is zero or PI. The vectors A,B are colinear
PERPENDICULAR
A& B are perpendicular to each other
Both are perpendicular
1 unit
Ortoghonal(90`)
If the sum and difference of two non-zero vectors A and B are equal in magnitude, then we can conclude that the magnitude of A is equal to the magnitude of B.
Let's assume that the magnitude of A and B is A=B=1 unit.
Now, we know that the sum of two vectors is given by A+B and the difference of two vectors is given by A-B.
Since the magnitude of A and B is the same, we can write the magnitude of the sum and the difference in terms of A as follows:
Magnitude of A+B = √(A^2 + B^2 + 2AB cosθ) = √(2 + 2cosθ)
Magnitude of A-B = √(A^2 + B^2 - 2AB cosθ) = √(2 - 2cosθ)
We know that the magnitude of the sum and the difference are equal, so we can set them equal to each other:
√(2 + 2cosθ) = √(2 - 2cosθ)
Squaring both sides, we get:
2 + 2cosθ = 2 - 2cosθ
Simplifying, we get:
cosθ = 0
This means that the angle between A and B is 90 degrees, so they are perpendicular to each other. Therefore, we can conclude that both A and B are perpendicular to each other, and their magnitude is 1 unit.