Find Dot Product Between Vector CD & Vector K

In summary, the dot product between vector CD and vector K is 0, and vector K is $\left<-2,-3\right>$. This can be found by knowing that the slope of vector K is perpendicular to the slope of vector A and that the magnitude of vector K is the same as vector A. Alternatively, we can use the dot product formula and solve for the x and y components of vector K, resulting in $\left<-2,-3\right>$.
  • #1
sp3
8
0
Hi! I'm given 2 points C(2;6) and D(0;10), a vector A with its components = (-3, 2). I'm asked to find the dot product between vector CD and an unknown vector K, knowing that K is perpendicular to A, same norm as A and with a negative x-component. I know that perpendicular means the dot product=0 and vector CD has a norm \sqrt{40} if i calculate it, but I have no clue how to solve it (we can't have a calculator).

Thank you for your help!
 
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  • #2
vector K would be $\left<-2,-3\right>$

vector CD would be $\left<-2,4\right>$

can you find the dot product?
 
  • #3
Thanks for the reply, how did you find vector K?
 
  • #4
sp3 said:
Thanks for the reply, how did you find vector K?

two ways ...

1. $\vec{A} = \left<-3,2 \right>$ has slope $m = -\dfrac{2}{3} \implies \vec{K}$ has slope $m_{\perp} = \dfrac{3}{2}$.

since $\vec{K}$ has a negative x component, then so does its y-component ... same magnitude means $\vec{K} = \left<x,y \right> = \left<-2,-3 \right>$

2. let $\vec{K} = \left<x,y\right>$

$\vec{A} \cdot \vec{K} = -3x + 2y = 0 \implies y = \dfrac{3}{2} x$

$|\vec{A}| = |\vec{K}| \implies \sqrt{(-3)^2 + 2^2} = \sqrt{x^2+y^2} \implies x^2+y^2 = 13 \implies x^2 + \dfrac{9}{4} x^2 = 13 \implies \dfrac{13}{4} x^2 = 13 \implies x = \pm 2$

$x < 0 \implies x = -2 \implies y = -3$
 
  • #5
Equivalently, one vector perpendicular perpendicular to (a, b) with the same norm is (-b, a), another is (b, -a). Here, K= (-3, 2) so those two perpendicular vector are (-2, 4) and (2, -3). The first has x component negative.
 

FAQ: Find Dot Product Between Vector CD & Vector K

1. What is the dot product between two vectors?

The dot product between two vectors is a mathematical operation that results in a scalar value. It is calculated by multiplying the corresponding components of the two vectors and then summing up the products. The dot product is also known as the scalar product or inner product.

2. How do I find the dot product between two vectors?

To find the dot product between two vectors, you need to multiply the corresponding components of the two vectors and then sum up the products. For example, if vector A = [a1, a2, a3] and vector B = [b1, b2, b3], the dot product would be a1*b1 + a2*b2 + a3*b3. You can use this formula for any number of dimensions.

3. What is the significance of the dot product in vector operations?

The dot product has many applications in vector operations. It can be used to calculate the angle between two vectors, determine if two vectors are perpendicular, and find the projection of one vector onto another. It is also used in physics and engineering to calculate work, force, and energy.

4. Can the dot product between two vectors be negative?

Yes, the dot product between two vectors can be negative. This can happen when the angle between the two vectors is greater than 90 degrees. In this case, the dot product represents the negative of the magnitude of the projection of one vector onto the other.

5. How is the dot product related to the magnitude of a vector?

The dot product between a vector and itself is equal to the square of its magnitude. This means that if vector A has a magnitude of ||A||, then the dot product of A with itself is A∙A = ||A||^2. This relationship is often used to calculate the magnitude of a vector when only its components are known.

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