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Definition. Vector parallel to one line or lying on one line are called collinear vectors (Fig. 1).
Condition of vectors collinearityTwo vectors are collinear, if any of these conditions done:Condition of vectors collinearity 1. Two vectors a and b are collinear if there exists a number n such that
a = n · b Condition of vectors collinearity 2. Two vectors are collinear if relations of their coordinates are equal.
N.B. Condition 2 is not valid if one of the components of the vector is zero. Condition of vectors collinearity 3. Two vectors are collinear if their cross product is equal to the zero vector.
N.B. Condition 3 applies only to threedimensional (spatial) problems. The proof of the condition of collinearity 3 Let there are two collinear vectors a = {a_{x}; a_{y}; a_{z}} and b = {na_{x}; na_{y}; na_{z}}. We find their cross product
= i (a_{y}na_{z}  a_{z}na_{y})  j (a_{x}na_{z}  a_{z}na_{x}) + k (a_{x}na_{y}  a_{y}na_{x}) = 0i + 0j + 0k = 0 Examples of tasksExamples of plane tasksExample 1. Which of the vectors a = {1; 2}, b = {4; 8}, c = {5; 9} are collinear?
Solution: Since the vectors does not contain a components equal to zero, then use the condition of collinearity 2, which in the case of the plane problem for vectors a and b will view:
Means:
Example 2. Prove that the vector a = {0; 3} and b = {0; 6} are collinear.
Solution: Since the vector components contain zero, then use the condition of collinearity 1, we find there is a number n for which: b = na.
For this we find a nonzero component of vector a in this case this is a_{y}. If the vectors are collinear then
Calculate the value of na: na = {2 · 0; 2 · 3} = {0; 6}Since b = 2a, the vectors a and b are collinear. Example 3. Find the value of n at which the vectors a = {3; 2} and b = {9; n} are collinear.
Solution: Since the vectors does not contain a components equal to zero, then use the condition of collinearity 2
Means:
Solve this equation:
Answer: vectors a and b are collinear when n = 6. Examples of spatial tasksExample 4. Which of the vectors a = {1; 2; 3}, b = {4; 8; 12}, c = {5; 10; 12} are collinear?
Solution: Since the vectors does not contain a components equal to zero, then use the condition of collinearity 2, which in the case of the plane problem for vectors a and b will view:
Means:
Example 5. Prove that the vector a = {0; 3; 1} and b = {0; 6; 2} are collinear.
Solution: Since the vector components contain zero, then use the condition of collinearity 1, we find there is a number n for which: b = na.
For this we find a nonzero component of vector a in this case this is a_{y}. If the vectors are collinear then
Calculate the value of na: na = {2 · 0; 2 · 3; 2 · 1} = {0; 6; 2}Since b = 2a, the vectors a and b are collinear. Example 6. Find the value of n and m at which the vectors a = {3; 2; m} and b = {9; n; 12} are collinear.
Solution: Since the vectors does not contain a components equal to zero, then use the condition of collinearity 2
Means:
From this relations we obtain two equations:
Solve this equations:
Answer: vectors a and b are collinear when n = 6 and m = 4.
Vectors
Vectors Definition. Main information
Component form of a vector with initial point and terminal point
Length of a vector
Direction cosines of a vector
Equal vectors
Orthogonal vectors
Collinear vectors
Coplanar vectors
Angle between two vectors
Vector projection
Addition and subtraction of vectors
Scalarvector multiplication
Dot product of two vectors
Cross product of two vectors (vector product)
Scalar triple product (mixed product)
Linearly dependent and linearly independent vectors
Decomposition of the vector in the basis
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