If the sum of two unit vectors is a unit vector, then the...

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is:
(a) $\sqrt{2}$
(b) $\sqrt{3}$
(c) $1 / \sqrt{2}$
(d) $\sqrt{5}$
240. What should be the angle between $\Delta \vec{A}$ and $\vec{A}$ so that $|\Delta \vec{A}|=\Delta|\vec{A}| ?$

Key Concepts

Differential Changes in Vector Magnitude

This concept involves examining how small changes to a vector, represented by ?A, affect the vector's magnitude, |A|. It is important in calculus and physics to know under what conditions the magnitude of the change vector equals the change in the magnitude of the vector. One key understandings here is that when the change is in the same (or exactly opposite) direction as the original vector, these quantities are equal.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is defined as the product of the magnitudes of the vectors and the cosine of the angle between them. This concept is crucial when determining the angle between vectors as well as in calculating the magnitudes of vector sums and differences.

Magnitude of Vector Difference

The magnitude of the difference between two vectors relates to both their individual magnitudes and the cosine of the angle between them. This is often computed using the law of cosines. In problems where the magnitudes of the vectors are known, understanding how changes in the angle affect the difference’s magnitude is essential.

Unit Vectors

Unit vectors are vectors with a magnitude of one. They are fundamental in representing direction in space and are often used in constructing other vectors through scalar multiplication and addition. Understanding unit vectors is key to exploring how different vectors combine and interact.

Vector Addition

Vector addition involves combining two or more vectors to produce a resultant vector. When adding unit vectors, their directions significantly influence the magnitude of the sum. The process uses geometric or algebraic methods, including the law of cosines, to determine the magnitude and direction of the resulting vector.

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