How to Find Magnitude of Vectors?

A vector is an object that has both magnitude and direction. To find the magnitude of a vector, we need to calculate the length of the vector. Let's talk more about the magnitude of a vector in this post.

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.

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Step by step guide to 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:

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

Magnitude of Vectors – Example 1:

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

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

\(∥ 𝑣⃗∥\)\(=\sqrt{3^2+5^2}\)

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

\(∥ 𝑣⃗∥\) \(=\sqrt{34}≅5.83\)

Magnitude of Vectors – Example 2:

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

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

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

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

\(∥ 𝑣⃗∥\) \(=\sqrt{50}=5\sqrt{2}\)

Exercises for Magnitude of Vectors

Find the magnitude of the vector.

  1. \(\color{blue}{𝑣⃗ =5i-4k+3j}\)
  2. \(\color{blue}{𝑣⃗ =6,8}\)
  3. \(\color{blue}{𝑣⃗ =4i-3k+2j-5l}\)
  4. \(\color{blue}{𝑣⃗ =4,-3}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{∥ 𝑣⃗∥}\)\(\color{blue}{=5\sqrt{2}}\)
  2. \(\color{blue}{∥ 𝑣⃗∥}\)\(\color{blue}{=10}\)
  3. \(\color{blue}{∥ 𝑣⃗∥}\)\(\color{blue}{=3\sqrt{6}}\)
  4. \(\color{blue}{∥ 𝑣⃗∥}\)\(\color{blue}{=5}\)

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