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Vectors in the coordinate system.
Operations with vectors given
with coordinates (reminder)
Operations with vectors given
with coordinates (reminder)
To discuss the basic relations and methods of coordinate geometry we
will need what we have learnt so far about vectors in a coordinate system.
will need what we have learnt so far about vectors in a coordinate system.
Position vector, base vectors, coordinates of a vector
DEFINITION: In the Cartesian coordinate system the position vector of the point \latex{P (x; y) } is the vector pointing from the origin to the point. (Figure 1)
\latex{P(x;y)}
\latex{\overrightarrow{p}}
\latex{ 1 }
\latex{ x }
\latex{ 0 }
\latex{ 1 }
\latex{ y }
Figure 1
THEOREM: If i is the position vector of the point (\latex{ 1; 0 }), and j is the position vector of the point (\latex{ 0; 1 }), then any vector \latex{\overrightarrow{a}} of the plane can unambiguously be expressed in the form \latex{\overrightarrow{a}=x_{1}i+a_{2}j} (as the linear combination of the vectors \latex{ i } and \latex{ j }). (Figure 2)
\latex{ i } and \latex{ j } are the base vectors of the coordinate system, and the real numbers \latex{a_{1}} and \latex{a_{2}} are the coordinates of the vector \latex{\overrightarrow{a}}. Notation: \latex{\overrightarrow{a} (a_{1};a_{2} )}
The coordinates of a vector in the coordinate system are the same as the coordinates of the end-point of its representative the starting point of which is the origin. It implies that in the coordinate system the coordinates of a given point and the coordinates of the position vector of this point are the same.
\latex{a_{2}\times j}
\latex{a_{1}\times i}
\latex{\overrightarrow{a}}
\latex{ 1 }
\latex{ 0 }
\latex{ 1 }
\latex{ j }
\latex{ i }
\latex{ x }
\latex{ y }
Figure 2
Operations with vectors given with coordinates
1. The coordinates of the sum of two vectors
If \latex{\overrightarrow{a}=a_{1}i+a_{2}j} and \latex{\overrightarrow{b}=b_{1}i+b_{2}j} then
\latex{\overrightarrow{a}+\overrightarrow{b}=(a_{1}+b_{1} )i+(a_{2}+b_{2} )j,}
i.e. the corresponding coordinates of the sum vector are obtained as the
sum of the corresponding coordinates of the vectors to be added. (Figure 3)
sum of the corresponding coordinates of the vectors to be added. (Figure 3)
\latex{b_{2}\times j}
\latex{a_{2}\times j}
\latex{a_{1}\times i}
\latex{b_{1}\times i}
\latex{a_{1}\times i}
\latex{b_{2}\times j}
\latex{\overrightarrow{a}}
\latex{\overrightarrow{a}+\overrightarrow{b}}
\latex{\overrightarrow{b}}
\latex{ 1 }
\latex{ 0 }
\latex{ 1 }
\latex{ y }
\latex{ j }
\latex{ i }
\latex{ x }
Figure 3
2. The coordinates of the difference of two vectors
If \latex{\overrightarrow{a}=a_{1}i+a_{2}j} and \latex{\overrightarrow{b}=b_{1}i+b_{2}j,} then
\latex{\overrightarrow{a}-\overrightarrow{b}=(a_{1}-b_{1} )i+(a_{2}-b_{2} )j,}
i.e. the corresponding coordinates of the difference vector are obtained as the difference of the corresponding coordinates of the two vectors. (Figure 4)
\latex{b_{2}\times j}
\latex{b_{1}\times i}
\latex{a_{1}\times i}
\latex{a_{2}\times j}
\latex{\overrightarrow{a}}
\latex{(a_{1}-b_{1} )\times i}
\latex{\overrightarrow{a}-\overrightarrow{b}}
\latex{(a_{2}-b_{2} )\times j}
\latex{\overrightarrow{a}-\overrightarrow{b}}
\latex{ 1 }
\latex{ 0 }
\latex{ 1 }
\latex{ y }
\latex{ x }
\latex{ j }
\latex{ i }
Figure 4
3. The coordinates of the scalar multiple of a vector
If \latex{\overrightarrow{a}=a_{1}i+a_{2}j} and \latex{\alpha} is a real number, then
\latex{\alpha \times \overrightarrow{a}= (\alpha \times a_{1} )i+(\alpha \times a_{2} )j,}
i.e. the coordinates of \latex{\alpha} times the vector are \latex{\alpha} times the coordinates.
The identities below are valid for the multiplication of vectors by real numbers (scalar multiplication):
\latex{\alpha \times \overrightarrow{a}+\beta \times \overrightarrow{a}=(\alpha +\beta )\times \overrightarrow{a},}
\latex{\alpha \times (\beta \times \overrightarrow{a} )=(\alpha \times \beta )\times \overrightarrow{a},}
\latex{\alpha \times (\overrightarrow{a}+\overrightarrow{b} )=\alpha \times \overrightarrow{a}+\alpha \times \overrightarrow{b}.}
\latex{\alpha \times (\beta \times \overrightarrow{a} )=(\alpha \times \beta )\times \overrightarrow{a},}
\latex{\alpha \times (\overrightarrow{a}+\overrightarrow{b} )=\alpha \times \overrightarrow{a}+\alpha \times \overrightarrow{b}.}
\latex{a_{2}\times j}
\latex{-a_{1}\times i}
\latex{-\overrightarrow{a}(-a_{1};-a_{2} )}
\latex{\overrightarrow{a}(a_{1};-a_{2} )}
\latex{a_{1}\times i}
\latex{-a_{2}\times j}
\latex{ 1 }
\latex{ i }
\latex{ 0 }
\latex{ 1 }
\latex{ j }
\latex{ y }
\latex{ x }
Figure 5
4. The coordinates of the negative of a vector
The vector opposite to the vector \latex{\overrightarrow{a}(a_{1};a_{2} )} is the vector \latex{-\overrightarrow{a}(-a_{1};-a_{2} ),} i.e. the coordinates of this vector are \latex{ –1 } times the coordinates of the
original vector. (Figure 5)
original vector. (Figure 5)
5. The dot product of two vectors
The dot product of the vectors \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} is
\latex{\overrightarrow{a}\times \overrightarrow{b}=|\overrightarrow{a}|\times |\overrightarrow{b|}\times \cos \alpha ,}
where \latex{\alpha} is the measure of the angle included between the representatives of the vectors with a common starting point \latex{(0°\leq \alpha \leq 180°).}
6. The dot product expressed in terms of the coordinates
If \latex{\overrightarrow{a}=a_{1}i+a_{2}j} and \latex{\overrightarrow{b}=b_{1}i+b_{2}j} then
\latex{\overrightarrow{a}\times \overrightarrow{b}=a_{1}b_{1}+a_{2}b_{2},}
i.e. the dot product of the two vectors is the sum of the product of the
corresponding coordinates.
corresponding coordinates.
Exercises
{{exercise_number}}. Draw a coordinate system and draw the position vectors of the following points.
- \latex{A(1;1)}
- \latex{B(-1;2)}
- \latex{C(-3;-4)}
- \latex{D(0;-6)}
- \latex{E(-2;4)}
- \latex{F(3;-5)}
{{exercise_number}}. Give the position vectors of the points given in the previous exercise in terms of the base vectors \latex{ i } and \latex{ j } (as the linear combination of the vectors \latex{ i } and \latex{ j }).
{{exercise_number}}. The vectors \latex{\overrightarrow{a}(1;2),\overrightarrow{b}(0;5),\overrightarrow{c}(-2;3)\overrightarrow{d}(7;-9)} are given. Give the coordinates of the following vectors.
- \latex{\overrightarrow{a}+\overrightarrow{b}}
- \latex{\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}}
- \latex{\overrightarrow{d}-\overrightarrow{b}}
- \latex{\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}}
- \latex{2\times \overrightarrow{c}}
- \latex{-3\times \overrightarrow{d}}
- \latex{\frac{1}{3}\times \overrightarrow{a}}
- \latex{-\frac{3}{5}\times \overrightarrow{d}}
- \latex{3\times \overrightarrow{a}+2\times \overrightarrow{b}}
- \latex{3\times \overrightarrow{c}-5\times \overrightarrow{d}}
- \latex{2\times (-2\times \overrightarrow{a}+6\times \overrightarrow{d} )}
- \latex{-\frac{3}{4}\times (7\times \overrightarrow{c}-5\times \overrightarrow{a} )}
{{exercise_number}}. Calculate the coordinates of the vectors \latex{\overrightarrow{AB}} and \latex{\overrightarrow{BA}}, if
- \latex{A(0;1), B(3;2);}
- \latex{A(4;1),B(-1;6);}
- \latex{A(-2;-5),B(7;-10);}
- \latex{A(8;-7),B(-4;-5);}
{{exercise_number}}. Give the dot product of the vectors \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} if
\latex{\overrightarrow{a}(1;2)\overrightarrow{b}(4;1);}
- \latex{\overrightarrow{a}(5;-2)\overrightarrow{b}(3;0);}
- \latex{\overrightarrow{a}(-7;5)\overrightarrow{b}(4;-9);}
- \latex{\overrightarrow{a}(-6;-10)\overrightarrow{b}(3;-11);}
{{exercise_number}}. Calculate the value of \latex{ x }, if we know that the vectors \latex{\overrightarrow{a}(5;7)} and \latex{\overrightarrow{b}(4;-x)} are perpendicular to each other.
