How to Find Magnitude of Vectors?

A vector has a direction and a magnitude. The magnitude of a vector refers to the size or length of the vector.

Related Topics

• How to Find Vector Components
• Vectors Introduction
• How to Find Scalar Multiplication of Vectors

A step-by-step guide to the magnitude of vectors

• The length of a vector determines its magnitude. The magnitude of the vector a is represented by the letter \(∥a∥\). The magnitude of a vector is always a positive or zero number, meaning it cannot be a negative number.

• For a two-dimensional vector a\(=(a_1,a_2)\), the formula for its magnitude is:

\(\color{blue}{ ∥a∥}\) \(\color{blue}{ =\sqrt{a_1^2+a_2^2}}\)

• For a three-dimensional vector a\(=(a_1,a_2,a_3)\),the formula for its magnitude is:

\(\color{blue}{ ∥a∥}\) \(\color{blue}{ =\sqrt{a_1^2+a_2^2+a_3^2}}\)

• The formula for a vector’s magnitude can be applied to any set of dimensions. For example, if a\(=(a_1,a_2,a_3,a_4)\), is a four-dimensional vector, the magnitude formula is:

\(\color{blue}{ ∥a∥}\) \(\color{blue}{ =\sqrt{a_1^2+a_2^2+a_3^2+a_4^2}}\)

The Magnitude of Vectors – Example 1:

Find the magnitude of the vector with \(?⃗=(3,5)\).

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Use this formula to find the magnitude of the vector: \(\color{blue}{ ∥a∥}\) \(\color{blue}{ =\sqrt{a_1^2+a_2^2}}\)

\(∥ ?⃗∥\)\(=\sqrt{3^2+5^2}\)

\(=\sqrt{9+25}\)\(=\sqrt{34}\)

\(∥ ?⃗∥\) \(=\sqrt{34}≅5.83\)

The Magnitude of Vectors – Example 2:

Find the magnitude of the vector \( ?⃗ =3i+4j−5k\).

Use this formula to find the magnitude of the vector: \(\color{blue}{ ∥a∥}\) \(\color{blue}{ =\sqrt{a_1^2+a_2^2+a_3^2}}\)

\(∥ ?⃗∥\) \(=\sqrt{(3^2)+(4^2)+(-5^2)}\)

\(=\sqrt{(9+16+25)}=\sqrt{50}\)

\(∥ ?⃗∥\) \(=\sqrt{50}=5\sqrt{2}\)

Exercises for Magnitude of Vectors

Find the magnitude of the vector.

1. \(\color{blue}{?⃗ =5i-4k+3j}\)
2. \(\color{blue}{?⃗ =6,8}\)
3. \(\color{blue}{?⃗ =4i-3k+2j-5l}\)
4. \(\color{blue}{?⃗ =4,-3}\)

1. \(\color{blue}{∥ ?⃗∥}\)\(\color{blue}{=5\sqrt{2}}\)
2. \(\color{blue}{∥ ?⃗∥}\)\(\color{blue}{=10}\)
3. \(\color{blue}{∥ ?⃗∥}\)\(\color{blue}{=3\sqrt{6}}\)
4. \(\color{blue}{∥ ?⃗∥}\)\(\color{blue}{=5}\)