# Equal vectors

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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.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}.

a ≠ c - as their coordinates are not equal,

b ≠ c - as their coordinates are not equal.

**Solution:**

a ≠ c - as their coordinates are not equal,

b ≠ c - 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.

a

a

**Solution:**

a

_{x}= b_{x}= 1a

_{y}= b_{y}=> 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}.

a ≠ b - as their coordinates are not equal,

b ≠ c - as their coordinates are not equal.

**Solution:**

a ≠ b - as their coordinates are not equal,

b ≠ c - 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.

a

a

a

**Solution:**

a

_{x}= b_{x}= 1a

_{y}= b_{y}= 2a

_{z}= b_{z}=> 4 = 2n => n = 4/2 = 2**Answer:** when n = 2 vectors a and b are equal.

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
Scalar-vector 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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