How Do You Solve Exercises Involving 2D and 3D Vectors and Dot Product?

[Kalbaan]

Hey!

I just joined the forum, but would like to get some help with 2D&3D vectors and dot product. I missed some classes due to a bad illness and now can't get the hang of it at all..
Would appreciate it alot, if someone could explain me how to solve these 5 exercises.
View attachment 1783

 

[Deveno]

Some hints:

a) Can you think of a way to use the dot product here?

b) If two vectors are of equal magnitude, but opposite direction, their vector sum is 0. Why is this relevant?

c) We have the 3 equations:

$3\lambda + 3\mu + 1 = x$

$-\lambda + 2\mu - 1 = y$

$4\lambda - 2 = z$

By multiplying equations (1) and (2) by suitable integers, can you eliminate $\mu$? Then try to use that equation and equation 3 to eliminate $\lambda$.

d) Such a line should be parallel to $v$, right?

e) Think about what the direction vectors of such a plane have to be...

 

[Kalbaan]

Thanks alot! You made my week mate!
Got the hang of them with your hints and my teachers powerpoint shows.

 

[Evgeny.Makarov]

There is another way of finding an equation of a plane in (c) and (e): an equation of the plane perpendicular to $(A,B,C)$ and passing through $(x_0.y_0,z_0)$ is $A(x-x_0)+B(y-y_0)+C(z-z_0)=0$, or $Ax+By+Cz+(-Ax_0-By_0-Cz_0)=0$.

 

[blue_raver22]

Hello,

Vectors and planes are important concepts in mathematics and physics. I am sorry to hear that you missed some classes due to illness, but I am happy to help you understand these concepts.

Firstly, let's start with 2D and 3D vectors. A vector is a quantity that has both magnitude (size) and direction. In 2D, vectors are represented by arrows on a coordinate plane, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction.

In 3D, vectors are represented by arrows in 3-dimensional space, where the length, direction, and orientation of the arrow represent the magnitude, direction, and orientation of the vector.

Now, let's talk about the dot product. The dot product is a mathematical operation that takes two vectors and produces a scalar (a single number) as a result. It is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them. In other words, it measures the amount of overlap or similarity between two vectors.

To solve the exercises, you will need to understand how to add, subtract, and multiply vectors. You can do this by breaking them down into their components (x, y, and z for 3D vectors) and using basic algebraic operations.

I would recommend reviewing your class notes or textbook for more detailed explanations and examples. Additionally, there are many online resources and videos that can help you understand these concepts better.

I hope this helps and good luck with your exercises!