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Table Of Contents

What are Equal Vectors?

Dot Product of Two Equal Vectors

Cross Product of Two Equal Vectors

Real-World Applications of Equal Vector

Common Mistakes and How to Avoid Them in Equal Vectors

Solved Examples of Equal Vector

FAQs on Equal Vector

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Last updated on September 26, 2025

Equal Vector

Q: What are equal vectors?

Equal vectors have identical length and point in the same direction.

Q: What is the formula for equal vectors?

The formula for equal vectors is that: if A = xi + yj + zk and B = qi + rj + sk, then A = B only if x = q, y = r, and z = s

Q: What is the angle between two equal vectors?

The angle between two equal vectors is 0 radians.

Q: What is the cross product of two equal vectors?

The cross product of any two equal vectors is 0, because sin(0°) = 0.

Q: Are all equal vectors parallel to each other?

Yes, the equal vectors are parallel to each other as they have the same direction.

Professor Greenline Explaining Math Concepts

Vectors are quantities that include both magnitude and direction. Two vectors are considered equal if they point in the same direction and have the same length. In this article, we will explore equal vectors, how to recognize them, and the key formulas and concepts related to them.

Equal Vector for US Students

What are Equal Vectors?

Two vectors are equal when they have the same magnitude and point in the same direction. In other words, vectors A and B are equal if both have the same direction and magnitude. This means they are collinear, have equal lengths, and point in the same direction. So, equal vectors are always parallel and have the same length and direction. But not all parallel vectors are equal.

Equations for Equal Vectors

Let’s consider two vectors, A = xi + yj + zk and B = qi + rj + sk, if x = q, y = r, and z = s, then the vectors A and B are equal.

For example, if two vectors are given as A = -4i + 5j + 7k and B = ai + bj + ck, find the values of a, b, and c if A = B
If vectors A and B are equal vectors, then corresponding components are equal.
So, a = -4
b = 5
c = 7

Equal Vectors Angle

Equal vectors have the same magnitude and direction, so they are parallel and same-directed. Thus, the angle between two equal vectors is always zero radians. Let’s prove it using the dot product formula. Consider two vectors P = ai + bj and Q = ai + bj .
Let the angle between them be θ
The dot product of vectors: P Q = |P||Q| cos
P Q = (ai + bj) ai + bj
= a2 + b2
|P|=|Q| = a2 + b2
cos = a2 + b2a2 + b2 a2 + b2
= a2 + b2a2 + b2 = 1
So, = cos-1(1) = 0

Dot Product of Two Equal Vectors

To multiply two vectors, we use a method called dot product, where the product is a scalar. The dot product of two vectors can be written as a b. The formula is: a b = |a| |b| cos ()
Where:
a is the magnitude (length) of vector a
b is the magnitude of vector b
is the angle between the two vectors

When the vectors are equal, then |a| =|b| and = 0, then a b = |a| |b| cos () becomes a a = |a|2cos(0)
a a = |a|21
a a = |a|2

Thus, the dot product of equal vectors are the squares of its magnitude.

For example, find the dot product of equal vectors u and v, where u = (2, -1, 4) and v = (2, -1, 4).
Since u and v are equal vectors, their dot product is the square of its magnitude
|u| = 22 + (-1)2 + 42
|u| = 4 + 1 + 16
|u| = 21
|u|2 = 212
= 21

Cross Product of Two Equal Vectors

The cross product is a method of multiplying two vectors that results in a new vector. It is written as a × b. The formula is:
a × b = |a||b|sin() n
Where:
a and b are the magnitudes of the two vectors
is the angle between them
n is a unit vector perpendicular to both a and b, indicating direction.

When two vectors are equal, the angle between them is 0°. Since sin(0°) = 0, the cross product becomes:
a × b = |a||b| sin(0o)n = 0
So, the cross product of two equal vectors is always zero vector.

Real-World Applications of Equal Vector

Equal vectors are used in various academic fields, such as mathematics, physics, and computer science. In this section, we will explore various applications of equal vectors.

In aviation, equal vectors help represent planes moving at the same speed and in the same direction. Since their velocity vectors are equal, this can be used to track flight motion and avoid collisions.
In computer graphics, vectors help show where objects are and which way they’re facing in 3D space. To make sure objects move or face in the same direction, we use equal vectors.
During a handoff in a relay race, the runner holding the baton and the runner receiving the baton must ensure their velocity vectors are equal to match their speed and direction. This helps execute an efficient transition that increases the chances of a win.

Common Mistakes and How to Avoid Them in Equal Vectors

When working with equal vectors, students often make errors due to the confusion between different types of vectors. In this section, we will learn some common mistakes and the ways to avoid them in equal vectors.

Mistake 1

Assuming vectors are equal without verifying

Students assume that the vectors are equal without comparing all components. So, always check if both the vectors have the same magnitude or direction, because vectors are said to be equal if they have the same magnitude and direction.

Mistake 2

Ignoring direction when comparing vectors

Some students only compare the magnitudes of vectors and forget to check their directions, which leads to mistakes. For example, a = (1, 2) and b = (-1, -2) have equal magnitudes, but opposite directions, which means they are not equal. Always check both magnitude and direction when comparing vectors.

Mistake 3

Misusing Components When Comparing Vectors

Sometimes, students might assume that two vectors with similar numbers are equal without considering the signs or individual components. For example, a = [3, -2] and b = [-3, 2] are not equal, but some may wrongly assume they are equal. For A to be equal to B, ensure that Ax = Bx and Ay = By.

Mistake 4

Errors while performing the dot product of equal vectors

When performing the dot product of equal vectors, students tend to make errors thinking that the dot product of equal vectors is the sum of the components, which is wrong. For example, students assume that the dot product of a = (2, -1, 4) and b = (2, -1, 4) is 2 + -1 + 4 = 5 instead of 21. To avoid this confusion, always remember that the dot product of equal vectors is the square of the magnitude of the vector, that is a b = |a|2.

Mistake 5

Incorrect cross-product assumptions

Students assume that two vectors are equal if their cross product is zero, but it is a wrong assumption. To check whether the vectors are equal, compare their components.