Tutorials › Calculus III › Equality of Vectors Component by Component

Home  /  Tutorials  /  Vector Components and Operations

Vector Components and Operations · Tutorial 30 of 999

Equality of Vectors Component by Component

Use corresponding components to decide when vectors are equal and to solve or check unknowns in vector equality statements.

Beginner 9 min read

What You'll Learn

  • State the component-by-component criterion for equality of vectors in three dimensions.
  • Compare vector components in their given order.
  • Solve vector equality statements for unknown components.
  • Check that all component equations are satisfied by the same values.
  • Identify and correct common mistakes when testing vector equality.

Why Vector Equality Is Component by Component

A vector in three-dimensional space can be represented by an ordered triple of components. These components record the vector's signed changes in the coordinate directions. When comparing two vectors, the order and value of each component matter: the first component is compared with the first, the second with the second, and the third with the third.

This gives a direct way to determine whether two vectors represent the same directed displacement. It also gives a systematic way to solve vector equalities containing unknown numbers: match corresponding components and solve the resulting scalar equations.

Definition: Two vectors written as ordered triples are equal if and only if their corresponding components are equal. Thus, for real numbers \(a,b,c,d,e,f\), $$ (a,b,c)=(d,e,f) \quad\Longleftrightarrow\quad a=d,\qquad b=e,\qquad c=f. $$

The phrase if and only if includes both directions. If the three corresponding components agree, the vectors are equal. If even one corresponding component differs, the vectors are not equal.

What the Criterion Means Geometrically

In the coordinate representation used for three-dimensional vectors, each component describes a signed change along one of the three coordinate directions. Two vectors have the same components precisely when they have the same signed change in every coordinate direction. Therefore, they represent the same vector, even if they are drawn with their tails at different points.

For instance, a vector with components \((2,-1,3)\) represents a change of 2 in the first coordinate, \(-1\) in the second, and 3 in the third. A vector based at another point but with those same components represents the same directed displacement. The location of a drawn arrow is not part of the ordered triple; its component values and their order are.

This criterion concerns equality of vectors, not equality of their positions or endpoints. In particular, checking only whether two arrows have the same starting point is not the test. The component comparison is the test.

A Short Justification

The component-by-component criterion follows from what an ordered triple means. The triple \((a,b,c)\) records three specified values in a specified order. Two such records are the same exactly when their first entries agree, their second entries agree, and their third entries agree.

Why the criterion works: If \((a,b,c)=(d,e,f)\), then the first coordinate of the two triples has one value, so \(a=d\). The same reasoning gives \(b=e\) from the second coordinate and \(c=f\) from the third. Conversely, if \(a=d\), \(b=e\), and \(c=f\), then the triples have identical entries in every position, so they are the same ordered triple. Thus the two conditions are equivalent.

For vectors, this ordered-triple fact applies to their component representations. It is not enough for the triples to contain the same numbers somewhere; each number must occur in the same position.

Worked Example: Testing Whether Two Vectors Are Equal

Worked Example: Testing Whether Two Vectors Are Equal

Determine whether \(\mathbf{u}=(4,-2,7)\) and \(\mathbf{v}=(4,-2,7)\) are equal.

Compare the first components: \(4=4\). Compare the second components: \(-2=-2\). Compare the third components: \(7=7\). All three corresponding pairs agree, so the vectors are equal:

$$ \mathbf{u}=\mathbf{v}. $$

Check: The two triples have the same first entry, the same second entry, and the same third entry. The equality criterion is satisfied in every component.

A comparison can also fail after just one component. There is no need for all three components to differ: one mismatch is enough to establish that the vectors are unequal.

Worked Example: Solving for Two Unknown Components

Worked Example: Solving for Two Unknown Components

Find \(x\) and \(y\) so that the following vector equality holds:

$$ (2x-1,\ y+3,\ 4)=(7,\ -2,\ 4). $$

Match corresponding components in order. The first components give \(2x-1=7\). Add 1 to both sides to get \(2x=8\), and divide both sides by 2 to obtain \(x=4\). The second components give \(y+3=-2\), so subtracting 3 from both sides gives \(y=-5\). The third components give \(4=4\), which is already true.

Therefore, \(x=4\) and \(y=-5\). Substitute these values into the left-hand triple:

$$ (2(4)-1,\ -5+3,\ 4) =(8-1,\ -2,\ 4) =(7,\ -2,\ 4). $$

Check: Substitution reproduces the right-hand triple component by component, so the values satisfy the original vector equality.

Notice that the third component still matters even though it contains no unknown. It confirms that the equality is consistent in that position. When solving, each coordinate supplies a condition, and every condition must hold.

Worked Example: Solving for Three Unknown Components

Worked Example: Solving for Three Unknown Components

Find \(a\), \(b\), and \(c\) so that

$$ (a+2,\ 3b-1,\ c/2)=(5,\ 8,\ -3). $$

Compare the first components: \(a+2=5\), so \(a=3\). Compare the second components: \(3b-1=8\). Adding 1 gives \(3b=9\), and dividing by 3 gives \(b=3\). Compare the third components: \(c/2=-3\). Multiplying both sides by 2 gives \(c=-6\).

Now substitute all three values into the left-hand side:

$$ (a+2,\ 3b-1,\ c/2) = (3+2,\ 3(3)-1,\ -6/2) = (5,\ 9-1,\ -3) = (5,\ 8,\ -3). $$

Check: The substituted triple agrees with the right-hand triple in all three positions. Thus \(a=3\), \(b=3\), and \(c=-6\) satisfy the equality.

All Component Conditions Must Be Compatible

When an unknown appears in more than one component, the same value must satisfy every equation produced by the comparison. Solving one component at a time does not mean the conditions can be treated as unrelated choices. They are all consequences of one proposed vector equality.

Worked Example: Checking Whether One Unknown Can Satisfy Every Component

Determine whether there is a real number \(t\) for which

$$ (t+1,\ 2t,\ 0)=(4,\ 5,\ 0). $$

The first components require \(t+1=4\), so \(t=3\). The second components require \(2t=5\), so \(t=5/2\). These requirements conflict: a single real number cannot equal both \(3\) and \(5/2\). The third components agree, since \(0=0\), but that does not resolve the conflict in the first two components.

To check directly, using \(t=3\), as required by the first component, gives

$$ (t+1,\ 2t,\ 0)=(4,\ 6,\ 0), $$

which does not equal \((4,5,0)\). Using \(t=5/2\), as required by the second component, gives

$$ (t+1,\ 2t,\ 0)=(7/2,\ 5,\ 0), $$

which also does not equal \((4,5,0)\). Therefore, there is no value of \(t\) that makes the vector equality true.

Check: Each component equation was considered, and the proposed values fail to provide one common solution. Hence the original equality has no solution.

A Common Mistake: Treating the Entries as Unordered

A frequent error is to see the same entries in two triples and conclude that the vectors are equal, without checking their positions. This ignores the fact that an ordered triple is ordered.

Common mistake and correction: It is incorrect to claim that \((1,2,3)=(1,3,2)\) because both triples contain 1, 2, and 3. Compare corresponding positions instead: the first entries match, \(1=1\), but the second entries do not, since \(2\ne 3\). Therefore the triples, and the vectors they represent, are not equal.

Swapping two components changes which coordinate direction has which signed change. It can change the vector even though none of the individual numbers has changed. Keep the components in their original order when writing them and when forming the equations.

A Reliable Procedure

For an equality between vectors written in component form, use the following procedure. It makes explicit both the order of comparison and the need to check every condition.

1
Write the triples in order.
Keep the first, second, and third components in their stated positions.
2
Match corresponding components.
Set the first component equal to the first, the second equal to the second, and the third equal to the third.
3
Solve all resulting conditions.
Each component equation must be satisfied by the same unknown values.
4
Substitute to check.
Put the proposed values into the original triple and verify that every position matches.

If the triples contain no unknowns, the same procedure is simply a test: all three comparisons must succeed for the vectors to be equal. If there are unknowns, the procedure produces equations whose solutions must be checked in the original vector equality.

Why This Criterion Is Useful

Component comparison turns a question about vectors into precise scalar conditions. In applications, this can identify whether two descriptions specify the same displacement, or reveal which unknown coordinate changes make two vector descriptions agree. The test is simple, but it depends on careful attention to order and on checking every coordinate.

It is also useful to distinguish a true vector equality from a partial match. If two components agree and one does not, the vectors are still unequal. If one unknown is constrained in several components, all of the resulting equations must be consistent. These checks prevent a plausible-looking partial calculation from being mistaken for a complete solution.

Key takeaway: For vectors in three dimensions, equality means equality of all corresponding components in order. To solve an unknown vector equality, match the components, solve every resulting equation, and verify the values in the original triples.

Check Your Understanding

Answer each question using the component-by-component criterion.

  1. Are \((3,-1,5)\) and \((3,-1,5)\) equal? State the three component comparisons.
  2. Are \((0,4,-2)\) and \((0,-2,4)\) equal? Identify the first position where the comparison fails.
  3. Find \(p\) and \(q\) if \((p-2,\ q+1,\ 6)=(5,3,6)\). Check your values by substitution.
  4. Is there a real number \(r\) such that \((r,\ r+1,\ 0)=(2,4,0)\)? Compare the conditions on \(r\).
  5. Why is matching the numbers in two triples without regard to their positions an invalid test for vector equality?

Previous

Next

Back to Tutorials