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Collinear vectors

Definition.
Vector parallel to one line or lying on one line are called collinear vectors (Fig. 1).
Collinear vectors
Fig. 1

Condition of vectors collinearity

Two 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 three-dimensional (spatial) problems.


The proof of the condition of collinearity 3

Let there are two collinear vectors a = {ax; ay; az} and b = {nax; nay; naz}. We find their cross product

a × b =  ijk  = i (aybz - azby) - j (axbz - azbx) + k (axby - aybx) = 
 ax  ay  az 
 bx  by  bz 

= i (aynaz - aznay) - j (axnaz - aznax) + k (axnay - aynax) = 0i + 0j + 0k = 0


Examples of tasks

Examples of plane tasks

Example 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:

ax  =  ay .
bx by

Means:

Vectors a and b are collinear because   1  =  2 .
4 8
Vectors a and с are not collinear because   1  ≠  2 .
5 9
Vectors с and b are not collinear because   5  ≠  9 .
4 8
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 ay. If the vectors are collinear then

n =  by  =  6  = 2
ay 3

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

ax  =  ay .
bx by

Means:

3  =  2 .
9 n

Solve this equation:

n =  2 · 9  = 6
3

Answer: vectors a and b are collinear when n = 6.

Examples of spatial tasks

Example 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:

ax  =  ay  =  az .
bx by bz

Means:

Vectors a and b are collinear because   1  =  2  =  3 .
4 8 12
Vectors a and с are not collinear because   1  =  2  ≠  3 .
5 10 12
Vectors с and b are not collinear because   5  =  10  ≠  12 .
4 8 12
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 ay. If the vectors are collinear then

n =  by  =  6  = 2
ay 3

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

ax  =  ay  =  az .
bx by bz

Means:

3  =  2  =  m
9 n 12

From this relations we obtain two equations:

3  =  2
9 n
3  =  m
9 12

Solve this equations:

n =  2 · 9  = 6
3
m =  3 · 12  = 4
9

Answer: vectors a and b are collinear when n = 6 and m = 4.

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