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Equal vectors

Definition. Vectors a and b is an equal vectors if they are in the same or parallel lines, their directions are the same and the lengths are equal (Fig. 1).

Two vectors are equal if they are collinear, codirected and have the same length:

a = b, if a↑↑b and |a| = |b|.

Conditions of vectors equality.Vectors are equal if their coordinates are equal.
Equal vectors
Fig. 1


Examples of tasks

Examples of plane tasks

Example 1. Determine which of the vectors are equal a = {1; 2}, b = {1; 2}, c = {3; 2}.

Solution:

a = b - as their coordinates are equal,
ac - as their coordinates are not equal,
bc - as their coordinates are not equal.
Example 2. At what value of the parameter n the vectors a = {1; 8;} and b = {1; 2n} are equal.

Solution:

Verify the equality of components of vectors
ax = bx = 1
ay = by => 8 = 2n => n = 8/2 = 4

Answer: when n = 4 vectors a and b are equal.

Examples of spatial tasks

Example 3. Determine which of the vectors are equal a = {1; 2; 4}, b = {1; 2; 2}, c = {1; 2; 4}.

Solution:

a = c - as their coordinates are equal,
ab - as their coordinates are not equal,
bc - as their coordinates are not equal.
Example 4. At what value of the parameter n the vectors a = {1; 2; 4} and b = {1; 2; 2n} are equal.

Solution:

Verify the equality of components of vectors
ax = bx = 1
ay = by = 2
az = bz => 4 = 2n => n = 4/2 = 2

Answer: when n = 2 vectors a and b are equal.

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