Clocluzy Test №1 Coordinate Method Variant 1. Choose the correct statement. A) The length of the vector |ä = x+y+z

Magnit

22

Sure! Here are the detailed explanations for each question:

1. Choose the correct statement.
A) The length of the vector |ä = x+y+z is calculated by the formula;
B) each coordinate of the vector is equal to the sum of the corresponding coordinates of its start and end points;
C) a vector is called coordinate if its length is equal to one;
D) each coordinate of the midpoint of the segment is equal to half the sum of the corresponding coordinates;
E) every point has positive coordinates.

The correct statement here is D. According to the midpoint formula, the coordinates of the midpoint of a line segment are half the sum of the corresponding coordinates of its endpoints. This means that for a line segment with endpoints (x1, y1, z1) and (x2, y2, z2), the coordinates of the midpoint would be \(\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}, \frac{{z_1 + z_2}}{2}\right)\). Therefore, each coordinate of the midpoint is equal to half the sum of the corresponding coordinates.

2. How far is point A(2; -3; -5) from the plane Oxu?
a) 2;
b) 3;
c) √38;
d) 10;
e) 5.

To find the distance from a point to a plane, we can use the formula:

\[d = \frac{{\left| Ax_0 + By_0 + Cz_0 + D \right|}}{{\sqrt{{A^2 + B^2 + C^2}}}}\]

In this case, the equation of the plane Oxu is x = 0. Therefore, we can substitute the coordinates of point A(2, -3, -5) into the formula and calculate the distance. Plugging in the values, we get:

\[d = \frac{{\left| 0 \cdot 2 + 1 \cdot (-3) + 0 \cdot (-5) + 0 \right|}}{{\sqrt{{0^2 + 1^2 + 0^2}}}} = \frac{3}{1} = 3\]

So the correct answer is b) 3.

3. Given points A(5, 3, 2), B(3, 4, 1), C(1, 1, 3), and D(-2, -1, 2), find the length of vector AB.

To find the length of a vector, we can use the formula:

\[| \mathbf{v} | = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}}\]

In this case, the coordinates of points A and B are (5, 3, 2) and (3, 4, 1) respectively. Plugging in these values into the formula, we get:

\[| AB | = \sqrt{{(3 - 5)^2 + (4 - 3)^2 + (1 - 2)^2}} = \sqrt{{4 + 1 + 1}} = \sqrt{6}\]

So the length of vector AB is \(\sqrt{6}\).