What conditions must be satisfied by the vectors "u" and "v" for the following to be true?
a) |u + v| = |u - v|
vector "u" is perpendicular to vector "v"
b) |u + v| > |u - v|
0° ≤ θ ≤ 90°
c) |u + v| < |u - v|
90° < θ ≤ 180°
------- Can you please explain to me why these conditions are true? Why is it perpendicular for the first one? Why is less less than 90°, but greater than 0° for question "b"? Why is it less than 180°, but greater than 90° for question "c"?
No I don't understand what you did.
Look at resultants
I will call them L and R for left and right
a)
Slope of U = Uy/Ux
Slope of V = Vy/Vx = -1/slope of U if perpendicular = -Ux/Uy
so
Vy/Vx = - Ux/Uy
- Vx Ux = VyUy
Now U + V = (Ux+Vx)i + (Uy+Vy)j
and U - V = (Ux-Vx)i + (Uy-Vy)j
magnitude of U+V squared =
(Ux+Vx)^2 + (Uy+Vy)^2
= Ux^2 + 2 UxVx +Vx^2 +Uy^2+2 UyVy^2+Vy^2
magnitude of U-V squared =
(Ux-Vx)^2 + (Uy-Vy)^2
= Ux^2 [[[[[-2 UxVx ]]]] +Vx^2 etc.
SEE WHAT IS HAPPENING?
If UxVx= - UyVy
those middle terms disappear and the magnitudes squared are the same.
which means that + or - the square roots are the same which means the absolute values are the same
a) For the equation |u + v| = |u - v| to be true, vector "u" must be perpendicular to vector "v." This means that the dot product of "u" and "v" is zero, i.e., u · v = 0. Geometrically, if two vectors are perpendicular, the angle between them is 90 degrees.
b) For the inequality |u + v| > |u - v| to be true, the angle ("θ") between vectors "u" and "v" must be between 0 and 90 degrees (inclusive) or, mathematically, 0° ≤ θ ≤ 90°. Geometrically, this means that the sum of the lengths of the vectors "u" and "v" is greater than the difference between their lengths.
c) For the inequality |u + v| < |u - v| to be true, the angle ("θ") between vectors "u" and "v" must be between 90 and 180 degrees (inclusive), or mathematically, 90° < θ ≤ 180°. Geometrically, this means that the sum of the lengths of the vectors "u" and "v" is less than the difference between their lengths.
To understand these conditions, we need to consider the geometric interpretation of vector sums and differences, as well as the properties of vector magnitudes and angles.
a) For the equation |u + v| = |u - v| to be true, the vectors u and v must be perpendicular. Here's why:
When we add two vectors, the resulting vector represents the diagonal of a parallelogram formed by the two original vectors. Similarly, when we subtract one vector from another, the resulting vector represents the diagonal of a parallelogram formed by the original vectors with one of them reversed.
If the lengths of these parallelograms' diagonals are equal, it means the parallelograms are congruent, indicating that the sides of the parallelograms are equal in length. This implies that the vectors u and v are equal in magnitude (or length).
Now, for the magnitude of the vector sum |u + v| to be equal to the magnitude of the vector difference |u - v|, the angle between the vectors u and v must be 90 degrees. This means that u and v are perpendicular to each other.
b) For the equation |u + v| > |u - v| to be true, the angle (θ) between the vectors u and v must be greater than or equal to 0 degrees, but less than or equal to 90 degrees.
To see why, let's consider the triangle inequality theorem. It states that in a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. In our case, the vectors |u + v|, |u|, and |v| form a triangle.
When the angle θ between u and v is between 0 and 90 degrees, the lengths |u + v| and |u - v| correspond to the sum of the lengths of two sides of the triangle, which is greater than the length of the remaining side.
c) For the equation |u + v| < |u - v| to be true, the angle (θ) between the vectors u and v must be greater than 90 degrees, but less than or equal to 180 degrees.
Again, using the triangle inequality theorem, when the angle θ between u and v is between 90 and 180 degrees, the length |u + v| corresponds to the sum of the lengths of two sides of the triangle, which is less than the length of the remaining side |u - v|.
In summary, the conditions for these equations to hold involve the relationship between the angles formed by the vectors u and v. For equation a), u and v must be perpendicular (angle of 90 degrees). For equations b) and c), the angles must lie in specific ranges (0 to 90 degrees for b) and 90 to 180 degrees for c)).