Two vectors 𝐀 and 𝐁 have precisely equal magnitudes. For the...

Key Concepts

Vector Addition and Subtraction

Vector addition combines two vectors to yield a resultant vector that represents the cumulative effect, while vector subtraction determines the difference between two vectors. The magnitudes of the sum and difference of two vectors depend on both their individual magnitudes and the angle between them, making it essential to understand how these operations relate geometrically and algebraically.

Magnitude and the Law of Cosines

The magnitude of the sum or difference of two vectors can be computed using a formula analogous to the Law of Cosines. This involves the squared magnitudes of the original vectors and a term involving the cosine of the angle between them, allowing one to relate the geometrical configuration of the vectors to the resulting magnitude.

Dot Product and Angle Between Vectors

The dot product is a scalar quantity defined for two vectors, in which it incorporates the cosine of the angle between them. It provides a bridge between algebraic operations and geometric interpretations, making it a key tool for finding the angle between vectors and understanding how that angle affects the resultant magnitude when vectors are added or subtracted.

Ratio Comparisons in Vector Magnitudes

Comparing the magnitudes of the sum and difference of two vectors by a given scaling factor involves setting up an equation based on their computed magnitudes. This process requires algebraic manipulation of the expressions derived through the Law of Cosines and trigonometric identities, ultimately enabling one to solve for the specific angle between the vectors that satisfies the prescribed ratio.

Transcript

00:02 Okay, in this problem, we have two vectors.

00:06 We have vector a and vector b.

00:10 And we're told that when we add the two vectors together, we get some distance, and we subtract the vectors.

00:23 We get some other distance.

00:26 So that's d plus and d minus.

00:29 And we're told that d, when we add them, is n times bigger than d minus.

00:38 And the question is, what's the angle that makes that happen? now, normally with vectors, i would want to find x and y components, but because we don't know what that angle is, it's really hard to break it into normal right triangles.

00:58 And so really, the only rule that i know that helps us with this is the law of cosines.

01:04 And that's the distance of the opposite angle.

01:08 So if this is the angle, it's the distance of that line, is equal to a squared plus b squared minus 2ab cosine theta.

01:21 And notice when theta is 90, that cosine term goes to zero, and that turns into the pythagram theorem.

01:27 So the pythagrin theorem is just a subset of the law of cosines.

01:34 We know though that a is equal to b and so i'm going to use the capital letters that i was using earlier so d squared is equal to a squared technically the magnitude of a plus a squared because b is the same thing as a times 2 times a times a times the cosine of theta well obviously that is a squared also and so d squared is equal to 2 a squared times one minus cosine theta if we simplify that equation.

02:13 So now i'm going to go ahead and treat this as just d and i'm just going to call this one n times d or d is the smaller distance and d is the larger distance and so i know that and in fact instead of using lowercase let me use a capital d and a capital d...

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