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Equal vector is formed when a vector and another vector have the same magnitude and direction. In layman's terms, two or more vectors are equal if their lengths are the same and they all point in the same direction. In general, we can determine whether two vectors are equal by comparing their coordinates. If the coordinates of two or more vectors are all the same, the vectors are equal. As a result, if vector A and vector B have the exact coordinates, they are said to be equal vectors.
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Key takeaways: Equal vector, vectors, collinear, magnitude, parallel and codirected vectors, Equal magnitude, length
What Is An Equal Vector?
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If vector A and vector B have the same magnitude and are pointing in the same direction, they are equal vectors. Two or more vectors are said to be 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, and z - components as the other vector. It is not necessary for equal vectors to begin at the same point.
Read more: Multiplication of a vector by a scalar
Equal Vector Diagram
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The diagram below depicts an equal vector A to vector B. The two vectors are equal but not co-initial, meaning they do not begin at the same point. As a result, equal vectors do not have to have the same initial points. They are parallel and codirected vectors of equal magnitude. If two vectors have the same magnitude but act in opposite directions, they are not equal vectors.
When Are Two Vectors Equal?
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When two or more vectors have the same length and point in the same direction, they are equal. Any two or more vectors that are collinear, codirected, and have the same magnitude are equal.
Mathematically, two vectors A and B are equal if they satisfy the following conditions:
A = B (Vector A and B are equal)
If and only if
|A| = |B| (Equal magnitude)
and
A ↑↑ B (Same direction)
If two vectors are equal, their column vectors must be equal as well. In other words, if the coordinates of two or more vectors are equal, they are equal.
Consider the vectors A = (ax1, ay1) and B = (ax1, ay1) (bx1, by1). If both of these vectors are equal, then:
ay1 = by1 and ax1 = bx1.
Equal vectors can start and end at different points, but their magnitudes and orientation must be the same.
In the image below, for example, all of the vectors except AB are equal even though they do not overlap. AB differs from the others in that, while it has the same magnitude, it does not have the same direction.
How To Compare Two Vectors?
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A comparison of vectors is essentially a comparison of the magnitudes and directions of the vectors.
To learn more about vector equality, this section will first compare vectors in three different examples. We'll then go over some practice problems and their step-by-step solutions to gain a better understanding of the subject.
Solved Example
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Example 1: Take a look at the image below. To determine whether the two vectors, a and b, are equal, compare them.
Solution: Two vectors are defined as equal if and only if they have the same magnitude in the same direction. The figure shows that vectors a and b are parallel and pointing in the same direction, but their magnitudes are not equal. As a result, we can conclude that the vectors given are not equal.
a ≠ b
This translates to saying that a vector of length 5m cannot be equal to a vector of length 10m. Two quantities of varying magnitudes cannot be equal. Scalars follow the same rule.
Also Read:
Things to Remember
- Two or more vectors with the same magnitude and direction are defined as equal vectors.
- This means that if vector A and vector B have the same length and point in the same direction, they are said to be equal vectors.
- Equal vectors are vectors with equal coordinates (and the same sign). As a result, while equal vectors are parallel vectors, parallel vectors may not be equal vectors.
- Mathematically, two vectors A and B are equal if they satisfy the following conditions:
- A = B (Vector A and B are equal): If and only if |A| = |B| (Equal magnitude), and A ↑↑ B (Same direction)
Sample Questions
Ques: Take a look at the image below. Determine whether or not the two vectors, a and b, are equal. (5 marks)
Ans: Two vectors are defined as equal if and only if they have the same magnitude in the same direction. This example is slightly more complex than the first. The above figure shows that vectors a and b have the same magnitude but do not point in the same direction. As a result, we can conclude that the given vectors are not equal once more.
a ≠ b
This is the equivalent of two cars leaving the same location at the same time. One is travelling northeast at a speed of 50 mph. The other vehicle is travelling south at a speed of 50 mph. Both cars will have travelled 50 miles in one hour, but they will have arrived at different destinations. Similarly, the velocity of the figure's vectors a and b cannot be equal. That is, two vectors of equal magnitude but opposite directions cannot be equal.
Ques: Take a look at the image below. Determine whether or not the two vectors, a and b, are equal. (3 marks)
Ans: This is a perfectly simple example. Vectors a and b have the same magnitude, as shown in the preceding image. The two vectors are pointing in the same direction as well. As a result, the two vectors are equivalent.
a = b
These three examples demonstrate that two vectors are only equal if they have the same length and direction.
Ques: Determine the magnitude of the following two vectors: PQ, with an initial point of O = (2,5) and a final point of W = (5,2), and OW, with an initial point of P = (-4, 2) and a final point of W = (5,2). (3,6). Is it true that the two vectors are equal? (5 marks)
Ans: The distance formula can be used to calculate the magnitude of the given vector OW:
|OW| = √ (5 – 2)2 + (2 – 5)2
Simplifying provides us with:
|OW| = √ (3)2 + ( – 3)2
|OW| = √ 9 + 9
|OW| = √ 18
|OW| = √ 2*9
|OW| = √ 2*(3)2
|OW| = 3 √ 2 units
As a result, the magnitude of vector OW is about 4.242 units.
Now we can calculate the magnitude of the given vector PQ:
|PQ| = √ (3 – ( – 4))2 + (6 – 2)2
Simplifying provides us with:
|PQ| = √ (7)2 + (4)2
|PQ| = √ 49 + 16
|PQ| = √ 65 units
As a result, the magnitude of vector PQ is approximately 8.062 units. These two vectors are not equal because neither their magnitudes nor their directions are the same.
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