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.
bxby
Means:
Vectors a and b are collinear because 1 = 2.
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Vectors a and с are not collinear because 1 ≠ 2.
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Vectors с and b are not collinear because 5 ≠ 9.
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Step-by-step explanation:
