Equal Vector

A vector is said to be an equal vector to another vector if they both have the same magnitude and the same direction. In simple words, we can say that two or more vectors are said to be equal vectors if their length is the same and they all point in the same direction. Generally, we can check for equal vectors by comparing their coordinates. If all the coordinates of two or more vectors are the same, then they are equal vectors. Hence, vector A is said to be an equal vector to vector B if they have equal coordinates.

Further, in this article, we will explore the concept of equal vector, its definition, and its formula. We will also understand the equal vector diagram and the angle between equal vectors with the help of some solved examples for a better understanding of the concept.

A vector A is called an equal vector to vector B if they have the same magnitude and are pointed in the same direction. We can also say that two or more vectors are equal if they are co-directed (directed in the same direction), collinear (lie on the same line), and have the same magnitude (having the same length). This implies that equal vectors are also parallel vectors. We can also check for an equal vector if it has the same x, y, z - components as the components of the other vector. It is not necessary for equal vectors to start from the same point.

Given below is the diagram of an equal vector A to vector B. The two vectors are equal vectors but are not co-initial vectors, that is, they do not start from the same point. Hence, equal vectors need not have the same initial points. They are parallel and codirected vectors with equal magnitude. If we have two vectors with an equal magnitude but are anti-parallel (acting in opposite directions), then they are not equal vectors.

Equal vectors are defined as two or more vectors that have the same magnitude and the same direction. This implies that vector A is said to be an equal vector to vector B if they have the same length and are pointing in the same direction. Vectors with equal coordinates (with the same sign) are said to be equal vectors. Hence, we can say that equal vectors are parallel vectors but parallel vectors may not be equal vectors.

The formula to check for an equal vector is: If we have two vectors \(\overrightarrow{A} = x\hat{i} + y\hat{j} + z\hat{k}\) and \(\overrightarrow{B} = p\hat{i} + q\hat{j} + r\hat{k}\), then vectors A and B are equal if and only if x = p, y = q, and z = r, that is, they have equal coordinates. To understand this formula, let us consider an example:

Example: If two vectors A = 2i + 3j - 8k and B = xi - yj - 8k are equal vectors, then find the values of x and y.

Solution: Since A and B are equal vectors, therefore they must have the same components. Hence we have,

2 = x, 3 = -y, -8 = -8

⇒ x = 2 and y = -3

Therefore, the values of x and y are x = 2 and y = -3.

As we know that equal vectors are also parallel vectors and the angle between parallel vectors is equal to zero radians. Now, we will prove this using the scalar product formula of vectors. Consider two equal vectors A and B with coordinates (x, y), that is, A = xi + yj and B = xi + yj and let θ be the angle between them. Now, take the scalar product of the vectors A and B:

A.B = |A| |B| cos θ

⇒ cos θ = (A.B)/(|A| |B|)

⇒ cos θ = [(xi + yj)(xi + yj)]/[√(x² + y²)√(x² + y²)]

⇒ cos θ = [(xi + yj)(xi + yj)]/[√(x² + y²)√(x² + y²)]

⇒ θ = arccos [(x² + y²)/(x² + y²)] [Taking inverse cosine on both sides]

⇒ θ = arccos (1)

⇒ θ = 0

Hence, the angle between two equal vectors is equal to zero.

☛ Related Topics:

• Radians
• Types of Vectors
• Direction of Vector

Important Notes on Equal Vector

• If two vectors have the same magnitude and the same direction, they are said to be equal.
• The angle between equal vectors is equal to zero degrees.
• Vector A is an equal vector to vector B if they have the same coordinates.

What is an Equal Vector in Math?

A vector X is said to be an equal vector to another vector Y if they both have the same magnitude and the same direction. In simple words, we can say that equal vectors have the same length and are pointing in the same direction.

What is Formula for Equal Vector?

The formula for equal vectors is: Two vectors A = xi + yj + zk and B = pi + qj + rk are said to be equal vectors if and only if x = p, y = q, and z = r.

When are Two Vectors Said to Be Equal Vectors?

Two or more vectors are said to be equal vectors when they have the same magnitude (length) and act in the same direction.

What is the Angle Between Two Equal Vectors?

The angle between two equal vectors is equal to zero degrees as they are parallel and act in the same direction. Also, the dot product of two equal vectors is equal to 1, hence the angle is equal to zero.

What is the Dot Product of Two Equal Vectors?

The dot product of two equal vectors is equal to 1 as they have the same magnitude and direction. Since their scalar product is equal to one, this implies the angle between them is equal to zero.

What Condition Should Be Satisfied For Two Vectors to Be Equal Vectors?

For two vectors to be equal, they must have the same length and should point in the same direction. In other words, we can say that equal vectors have the same coordinates.

How to Determine If Two Vectors Are Equal?

To check if two vectors are equal, we check if their magnitudes are the same and they must be co-directed, that is, act in the same direction.