Question

Select the correct answers. which relationships hold true for the sum of the magnitudes of vectors u and v, which are perpendicular?

Answer 1

The 2nd relationship || u + v || = √ { || u ||² + || v ||² }, and the 4th relationship || u + v || < || u || + || v || holds for the sum of the magnitudes of vectors u and v, which are perpendicular.

In the question, we are asked to identify the relationships that hold for the sum of the magnitudes of vectors u and v, which are perpendicular to each other.

The magnitude of a vector A is shown as || A ||.

Thus, the magnitude of vector u is || u ||, and of vector, v is || v ||.

By vector algebra, we know that,

|| u + v || = √ { || u ||² + || v ||² + 2( || u || )( || v || ) cos θ }, where θ is the angle between vector u and vector v.

Now, we are given that vectors u and v are perpendicular to each other, thus, θ = 90°, which gives cos θ = 0, or,

|| u + v || = √ { || u ||² + || v ||² }, making the 2nd relation true.

Now, we have, || u + v || = √ { || u ||² + || v ||² }.

Squaring both sides, we get:

|| u + v || ² = || u ||² + || v ||² = { || u || + || v || }² - 2(|| u ||)(|| v ||),

or, || u + v || ² < { || u || + || v || }² {Since, 2(|| u ||)(|| v ||) > 0},

or, || u + v || < || u || + || v || {Taking square roots}, making the 4th relation true.

Thus, the 2nd relationship || u + v || = √ { || u ||² + || v ||² }, and the 4th relationship || u + v || < || u || + || v || holds for the sum of the magnitudes of vectors u and v, which are perpendicular.

Learn more about perpendicular vectors at

brainly.com/question/1370501

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For the complete question, refer to the attachment.