Finding a vector; need work checked

In summary, in order to find a vector of length 7 in the direction opposite to (-3,4)^T, one can simply take a scalar multiple of the vector <3,-4> and adjust the values accordingly. This can be done by dividing the components by 5 and then multiplying by 7, resulting in a vector with components of <21/5,-28/5>, which has a length of 7.
  • #1
Whitishcube
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0

Homework Statement


Find a vector of length 7 in the direction opposite to (-3,4)^T


Homework Equations





The Attempt at a Solution


So i start off by drawing the vector (-3,4)^T (the T power meaning transposed), and then drawing a vector in the opposite direction of length 7. The first vector has a length of 5. My method involved separating each vector into its components and making two triangles out of them, and then using ratios involving the two known components of the first vector (-3 and 4), the components of the second vector (dubbed v_1 and v_2), and the lengths of each vector (5 and 7). The answer I got was v_1 = (-21/5) and v_2=(28/5).

I have included a picture in the attatchment. Thank you.
 

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  • #2
You complicated this problem by over thinking.

if -a is opposite in direction to a, then what vector is opposite to -3i+4j ?

Then consider that any vector in this same direction will be a scalar multiple of this vector, you want its magnitude to be 7.
 
  • #3
I see what you mean haha. I'm new to the thinking in linear algebra. I got 21/5i and -28/5j for the vector. Is that correct?
 
  • #4
It's easy enough to check. Your vector should have a length of 7, and should be a multiple of <3, -4>.
 
  • #5
Yup. Checks out. Thanks!
 

Related to Finding a vector; need work checked

1. How do I find the magnitude of a vector?

To find the magnitude of a vector, you need to use the Pythagorean theorem. Simply square each component of the vector, add the squares together, and take the square root of the sum. This will give you the magnitude of the vector.

2. What is the difference between a scalar and a vector?

A scalar is a quantity that has only magnitude, while a vector has both magnitude and direction. Scalars are represented by a single number, while vectors are represented by magnitude and direction components.

3. How do I add two vectors together?

To add two vectors, you must add their corresponding components. For example, to add vectors (2,3) and (5,1), you would add 2+5 to get the x component and 3+1 to get the y component. The resulting vector would be (7,4).

4. Can a vector have a negative magnitude?

No, a vector's magnitude is always positive. However, a vector can have a negative direction if it is pointing in the opposite direction of a positive vector.

5. How do I find the direction of a vector?

To find the direction of a vector, you need to use inverse trigonometric functions. First, find the angle between the vector and the positive x-axis. Then, use the inverse tangent function to find the angle in radians or degrees.

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