Question

Даны точки М(-4;-2;1), Н(№;-1;-1), К(2;1;-3).Найдите: а)Координаты вектора МН и КМ; б)Модуль вектора МН; в) Координаты вектора ВС=3МН-2КМ

Answer 1

MH {7;1;-2}

KM {-6;-3;4}

 

|MH|=\/49+1+4=\/54=3\/6

\/ - корень

 

 3MH{21;3;-6}

-2KM{12;6;-8}

BC{33;9;2}

Answer 2

а) Координаты вектора МН и КМ

To find the coordinates of vector MN and KM, we can subtract the coordinates of point M from the coordinates of point N and K, respectively.

Let's start with vector MN: MN = N - M

The coordinates of point N are (-1, -1, -1), and the coordinates of point M are (-4, -2, 1).

Subtracting the corresponding coordinates, we get: MN = (-1 - (-4), -1 - (-2), -1 - 1) = (3, -1, -2)

So, the coordinates of vector MN are (3, -1, -2).

Now, let's find the coordinates of vector KM: KM = M - K

The coordinates of point K are (2, 1, -3), and the coordinates of point M are (-4, -2, 1).

Subtracting the corresponding coordinates, we get: KM = (-4 - 2, -2 - 1, 1 - (-3)) = (-6, -3, 4)

So, the coordinates of vector KM are (-6, -3, 4).

б) Модуль вектора МН

To find the modulus (or magnitude) of vector MN, we can use the formula:

|MN| = sqrt((x^2) + (y^2) + (z^2))

where (x, y, z) are the coordinates of vector MN.

In our case, the coordinates of vector MN are (3, -1, -2).

Substituting the values into the formula, we get: |MN| = sqrt((3^2) + (-1^2) + (-2^2)) = sqrt(9 + 1 + 4) = sqrt(14)

So, the modulus of vector MN is sqrt(14).

в) Координаты вектора ВС = 3МН - 2КМ

To find the coordinates of vector VS, we can use the formula:

VS = 3MN - 2KM

where MN and KM are the previously calculated vectors.

Substituting the values, we get: VS = 3(3, -1, -2) - 2(-6, -3, 4) = (9, -3, -6) - (-12, -6, 8) = (9 + 12, -3 + 6, -6 - 8) = (21, 3, -14)

So, the coordinates of vector VS are (21, 3, -14).