Vectors — Magnitude, Direction, Addition & Dot Product

A vector describes size and direction at the same time. In coordinates, a vector such as $v = (3, 4)$ or $v = (2, -1, 5)$ tells you how far it moves along each axis. From those components, you can find magnitude, add vectors, and compute a dot product.

If you only remember one idea, remember this: vectors are not just lengths. Direction is part of the quantity, so the arithmetic has to preserve direction too.

What vectors mean in coordinates

A scalar has size only. Temperature, mass, and time are common scalar examples. A vector has size and direction. Displacement, velocity, and force are standard examples.

In basic math and physics, vectors are often written as ordered lists of components. In $2$ dimensions,

$$v = (v_1, v_2)$$

and in $3$ dimensions,

$$v = (v_1, v_2, v_3).$$

The number of components matters. You can only add vectors directly, or take the standard dot product, when the vectors live in the same dimension.

How to find vector magnitude

The magnitude of a vector is its length. In the usual Euclidean setting, the magnitude of $v = (v_1, v_2)$ is

$$|v| = \sqrt\{v_1^2 + v_2^2\}$$

and for $v = (v_1, v_2, v_3)$ it is

$$|v| = \sqrt\{v_1^2 + v_2^2 + v_3^2\}.$$

This is the vector version of the Pythagorean idea. Magnitude tells you how long the vector is, while the signs and relative sizes of the components help determine its direction.

One useful caution: the zero vector has magnitude $0$, but it does not point in one unique direction.

How vector addition works

To add vectors, add matching components:

$$(a_1, a_2) + (b_1, b_2) = (a_1 + b_1, a_2 + b_2).$$

The result is another vector. That matters because the sum still has both size and direction.

This is why you usually cannot add only the magnitudes. If two vectors point in different directions, their combined effect depends on both directions, not just on how large the numbers are.

What the dot product tells you

The dot product takes two vectors of the same dimension and returns a scalar:

$$a \cdot b = a_1b_1 + a_2b_2 + \dots + a_nb_n.$$

This tells you how much the vectors line up. In the usual Euclidean setting, it also satisfies

$$a \cdot b = |a||b|\cos(\theta),$$

where $\theta$ is the angle between the vectors.

That formula gives a quick interpretation:

• If $a \cdot b > 0$, the angle is acute.
• If $a \cdot b = 0$, the nonzero vectors are perpendicular.
• If $a \cdot b < 0$, the angle is obtuse.

This angle interpretation depends on the usual Euclidean dot product. That is the standard version used in introductory math and physics.

Worked example: magnitude, addition, and dot product together

Let

$$a = (3, 4), \qquad b = (4, -3).$$

Start with magnitude. For $a$,

$$|a| = \sqrt\{3^2 + 4^2\} = \sqrt\{25\} = 5.$$

For $b$,

$$|b| = \sqrt\{4^2 + (-3)^2\} = \sqrt\{25\} = 5.$$

So both vectors have the same size, even though they point in different directions.

Now add them:

$$a + b = (3 + 4,\ 4 + (-3)) = (7, 1).$$

The sum is a new vector, not the number $10$. Its magnitude is

$$|a + b| = \sqrt\{7^2 + 1^2\} = \sqrt\{50\}.$$

Now compute the dot product:

$$a \cdot b = 3 \cdot 4 + 4 \cdot (-3) = 12 - 12 = 0.$$

Because the dot product is $0$, these nonzero vectors are perpendicular in the usual Euclidean plane. This one example shows the main pattern clearly:

• magnitude measures size
• addition creates a new vector
• the dot product measures alignment

Common vector mistakes

Adding magnitudes instead of vectors

Adding $|a| + |b|$ is not the same as finding $|a + b|$. Those are different quantities unless the vectors point in the same direction.

Ignoring the same-dimension condition

You cannot directly add a $2$D vector to a $3$D vector, and you cannot take the standard dot product between them either.

Mixing up dot product and multiplication by a number

The dot product gives one scalar. It does not produce another vector.

Using angle rules without the right setting

The formulas for magnitude and the geometric dot-product interpretation above assume the usual Euclidean setting. That is the standard setting in most introductory courses, but it is still a condition.

Where vectors are used

Vectors appear anywhere direction matters. In geometry, they help describe points, lines, projections, and angles. In physics, they are used for displacement, velocity, acceleration, and force. In engineering and graphics, they help represent motion, orientation, and changes in space.

You do not need advanced linear algebra to start using vectors well. For many problems, the whole job is just: write the components correctly, apply the right operation, and interpret the result.

Try a similar vector problem

Change the example to $a = (2, 1)$ and $b = (1, 2)$. Find the magnitude of each vector, add them, and compute the dot product. Then decide whether the angle between them is acute, right, or obtuse.

If you want a quick check, solve the same pair by hand first and then compare with a solver. That makes sign mistakes and component mix-ups much easier to catch.