Vectors

How to find the length of a 3D vector?

Quick answer

To determine the length of a 3D vector v = (x, y, z), square each component, add the three squares together, and take the square root of the sum. The formula is |v| = √(x² + y² + z²). For example, the length of (2, 3, 6) is √(4 + 9 + 36) = √49 = 7.

The topic explained

The 3D length formula

Determining the length of a 3D vector relies directly on extending the Pythagorean theorem into three dimensions. A vector in space has three components: x along the horizontal axis, y along the depth axis, and z along the vertical axis, written as v = (x, y, z) or xi + yj + zk.

The length or magnitude is written as |v|. Because the three axes meet at right angles, you find the diagonal across the base using x² + y², and then combine that with the vertical height z. This gives the formula |v| = √(x² + y² + z²). The result is always a non-negative number that represents pure distance.

Finding the vector between two points first

In geometric problems, you often start with two points instead of a single vector. If point A is (x₁, y₁, z₁) and point B is (x₂, y₂, z₂), the displacement vector AB is found by subtracting the starting coordinates from the ending coordinates: AB = (x₂ - x₁, y₂ - y₁, z₂ - z₁).

Once you have these components, apply the magnitude formula. For instance, to find the distance between A(1, 2, 1) and B(4, -2, 13), first find the vector AB = (4 - 1, -2 - 2, 13 - 1) = (3, -4, 12). Next, calculate the length: √(3² + (-4)² + 12²) = √(9 + 16 + 144) = √169 = 13.

Common pitfalls with negative signs

The most common mistake occurs when squaring negative numbers. A negative component squared must always yield a positive value. Writing -4² on a calculator without parentheses gives -16, which will produce an incorrect answer. Always treat (-4)² as 16.

Another check is geometry: the length of a vector can never be shorter than the absolute value of any individual component. If one coordinate difference is 12, the total length must be at least 12.

Free printable worksheet

3D Vector Length and Spatial Distance Practice

Designed for Grade 12 students. Calculate the component vector and its magnitude for each problem. Simplify all radicals where possible.

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Free printable worksheetlessoons.com

3D Vector Length and Spatial Distance PracticeName:

Find the length of the vector u = (0, 3, 4).
Find the magnitude of the vector v = (2, -3, 6).
Calculate the length of vector w = (1, 2, 2).
Find the displacement vector AB and its length for points A(1, 0, 3) and B(3, 4, 7).
Find the distance between point P(5, -1, 2) and point Q(2, 3, -10).
A box has one corner at the origin (0, 0, 0) and the opposite corner at (4, 7, 4). Find the length of the main interior diagonal.
Vector a has components (-2, 5, -4). Find its exact length.

Common questions

Can the length of a 3D vector ever be negative?

No. Length represents geometric distance, which is always zero or positive. Even if all three coordinates are negative, squaring them turns each into a positive number, ensuring the sum under the square root is non-negative.

What is the difference between vector magnitude and vector length?

They mean the exact same thing. 'Magnitude', 'length', and 'norm' are interchangeable mathematical terms describing how long a vector is from its initial point to its terminal point.

How do you turn a 3D vector into a unit vector?

First find the length of the vector using the standard formula. Then divide each individual component (x, y, and z) by that length. The resulting vector points in the same direction but has a length of exactly 1.

What topic usually comes after 3D vector lengths?

After calculating lengths and distances in space, students typically learn the dot product (scalar product). This allows you to determine whether two spatial vectors are perpendicular and calculate the angle between them inside 3D shapes.

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