How Can You Convert Points to Vectors and Calculate Median Lengths in Triangles?

[xairylle]

Hi, I'm Xairylle and I need a little help with this. I think it's kind of basic electromagnetics stuff but I don't know how this goes given that all I've got is a problem nothing else. I am so sorry, I didn't see this room in the forum earlier so I made a post in the General Engineering section. Is that a bad thing? If so, I am sorry.

1. Express each of the given points into its equivalent vector
a) A(2,1,-5)
b) A(1,4,6) B(5,-3,0)

2. Find the lengths of the medians of the given triangles ABC
a) A(2,1,3) B(3,-1,-2) C(0,2,-1)

I need solutions and answers since I really can't understand. Please help if it's not too much trouble. Thank you.

 

[Ouabache]

Welcome xairylle to the physics forums!
You will see there are lots of folks willing to help you, but first you need to show what you have tried. (You may have noticed this https://www.physicsforums.com/showthread.php?t=94388) at the top of this page. It would be useful to read through that).

 

[piercas]

Hi Xairylle,

I am happy to help you with your questions about vectors and points. It's great that you are seeking clarification and understanding, that's an important quality for a scientist!

1. Expressing points as vectors:
A vector is a mathematical representation of a quantity that has both magnitude and direction. In this case, we can express a point as a vector by considering its position relative to the origin (0,0,0). The vector will have three components, representing the x, y, and z coordinates of the point.

a) A(2,1,-5) can be expressed as a vector <2,1,-5>.
b) A(1,4,6) can be expressed as a vector <1,4,6> and B(5,-3,0) can be expressed as a vector <5,-3,0>.

2. Finding the lengths of medians of triangles:
The median of a triangle is a line segment connecting a vertex to the midpoint of the opposite side. To find the length of a median, we can use the distance formula: d = √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2).

a) In triangle ABC, the median from A to BC can be found by finding the midpoint of BC, which is ((3+0)/2, (-1+2)/2, (-2-1)/2) = (1, 0.5, -1.5). Then, using the distance formula, we can find the length of the median from A to BC as d = √((1-2)^2 + (0.5-1)^2 + (-1.5-3)^2) = √(1+0.25+16.25) = √17.5.

b) Similarly, we can find the length of the median from B to AC by finding the midpoint of AC, which is ((2+3)/2, (1+2)/2, (3-1)/2) = (2.5, 1.5, 1). Then, using the distance formula, we can find the length of the median from B to AC as d = √((2.5-3)^2 + (1.5-(-1))^2 + (1-(-2))^2) = √(0