Finding Two Vectors from Vector Sum / Difference

In summary, by adding the two given equations and subtracting them, we can find the values for vectors v and w, which are (3,3) and (2,-2) respectively.
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Cod

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Finding Two Vectors from Given Linear Combination

Homework Statement


If v + w = (5,1) and v - w = (1,5), compute and draw v and w.

Homework Equations


v + w = (5,1)
v - w = (1,5)

The Attempt at a Solution


I understand how to find the sum of two vectors, but I'm confused on how to find the vectors from a sum / difference. My first attempt was to draw the v + w vector and v - w vector and try to visualize the original two vectors, v and w. However, I couldn't come up with anything productive.


If it matters, I'm going through MIT OCW 18.06 w/ Introduction to Linear Algebra (Strang) and this is question 3 from problem set 1.1. Its been a few years since I took LA or any vector math and I'm trying to regain lost knowledge.

Any guidance would be greatly appreciated.
 
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  • #2
"I understand how to find the sum of two vectors, but I'm confused on how to find the vectors from a sum / difference."

What happens if you add the two equations together in Section 2 of the OP? What happens if you subtract them?
 
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  • #3
SteamKing said:
"I understand how to find the sum of two vectors, but I'm confused on how to find the vectors from a sum / difference."

What happens if you add the two equations together in Section 2 of the OP? What happens if you subtract them?
Adding them would give 2v = (6,6), which would make vector v = (3,3). Subtracting them would give 2w = (4,-4), which would make vector w = (2,-2).

Wow, wish I would've seen that on my own. Thanks for the "nudge" in the correct direction.
 

1. How do you find the magnitude of the resultant vector?

The magnitude of the resultant vector can be found by using the Pythagorean theorem, where the square of the magnitude is equal to the sum of the squares of the individual vector magnitudes. The square root of this value will give you the magnitude of the resultant vector.

2. Can you find the direction of the resultant vector?

Yes, the direction of the resultant vector can be found by using trigonometric functions such as sine and cosine. The direction can be calculated using the formula: tanθ = (Δy/Δx), where θ is the angle between the resultant vector and the horizontal axis.

3. What is the difference between the vector sum and vector difference?

The vector sum is the combination of two or more vectors, while the vector difference is the result of subtracting one vector from another. The resultant vector in the vector sum represents the overall effect of all the individual vectors, while the resultant vector in the vector difference represents the difference between the two vectors.

4. How do you find the components of a resultant vector?

The components of a resultant vector can be found by using basic trigonometry. The x-component can be calculated using the formula: Rx = R cosθ, where θ is the angle between the resultant vector and the horizontal axis. The y-component can be calculated using the formula: Ry = R sinθ, where θ is the angle between the resultant vector and the horizontal axis.

5. Can you use the parallelogram method to find the resultant vector?

Yes, the parallelogram method is one of the graphical methods used to find the resultant vector. By drawing the vectors to scale, forming a parallelogram, and drawing the diagonal, the magnitude and direction of the resultant vector can be determined using the length and angle of the diagonal.

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