Examples of solving problems to the section Linear algebra and analytic geometry

Example N 1

In PDK are given vectors Find the unit vector x perpendicular to the vectors a, b .

Decision.

The vector c = a × b is perpendicular to the vectors a, b .

As the desired vector x you can take the vector or vector

Vectors form the right three vectors, and - left.

Answer: or

Example N 2

To find if it is known that The angle φ between the vectors a, b is φ / 6.

Decision.

. Using the properties of the scalar product, we find

Answer:

Example N 3

Vectors are specified in PDK , , , . Prove that the vectors a, b, c form a basis, and find the expansion of the vector d in the basis of a, b, c .

Decision.

Find the determinant composed of the coordinates of the vectors a, b, c :

Because Δ ≠ 0, then a, b, c is a basis. Now we find the expansion of the vector d in the basis of a, b, c . It is necessary to find the numbers α ₁ , α ₂ , α ₃ such that a α ₁ + b α ₂ + c α ₃ = d . In expanded form, this equality is a linear system of algebraic equations with unknowns α ₁ , α ₂ , α ₃ :

According to Kramer’s formulas we find:

Answer:

Example N 4

Vectors are specified in PDAC , , . Find a vector x such that its scalar product with vectors a, b, c equals - 12, 6, - 8, respectively.

Decision.

Let be From the condition of the problem we get

It is necessary to solve a system of linear algebraic equations with unknowns x ₁ , x ₂ , x ₃ . According to Kramer’s formulas we find

Answer:

Example N 5

Vectors are specified in PDAC , . Find the area of ​​the parallelogram built on the vectors x, y where the vector x is perpendicular to the vectors a, b , and x = 3 , and the vector
y = a + b .

Decision.

The vector c = a × b is perpendicular to the vectors a, b

The vector x must be collinear to the vector c . Since the condition of the problem is x = 3 , then Find ,

area parallelogram built on vectors x, y .

Answer:

Example N 6

Vectors are specified in PDAC , and points , . Find the volume of a parallelepiped constructed on vectors Where

Decision.

- volume of parallelepiped built on vectors .

Answer:

Example N 7

At what value of x will the vectors a, b, c be coplanar if

Decision.

Vectors a, b, c will be coplanar if ( a, b, c ) = 0 .

Answer: x = 8.

Example N 8

In the triangle with vertices K (-5; 4), L (1; -4), M (-9; 1) find:
a) the equation of a straight line containing the height dropped from the vertex L ;
b) the length of the height lowered from the vertex L ;
c) the point N , symmetric to the point L , with respect to the straight line passing through the points K, M ;
d) the equation of a line containing the bisector of the angle L.

Decision.

a) the normal vector of a straight line (Lh) containing height is a vector Find the equation of the line

- the equation of a straight line containing the height lowered from the top;

b) the length of the height dropped from the vertex L is equal to the distance ρ from the point L to the straight line (K, M) passing through the points K, M.

Find the equation of this line. Because that is the normal vector of this line. Find the general equation of the line (K, M) :

The normal equation of the line (K, M) is

- height length;

c) to find the point N you need to determine the point of intersection direct (Lh), (KM) i.e. it is necessary to solve a system of linear equations

Now we find the point N (x; y): .

d) vector is a straight line vector (LK) , and vector - direct (LM) . Let be where s _x , s _y - unknown vector of a line (Ll) containing the bisector of the angle L. Vector s forms with vectors equal angles. Consequently, Because

Answer: