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.
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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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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 \(?⃗=(3,5)\).
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.
- \(\color{blue}{?⃗ =5i-4k+3j}\)
- \(\color{blue}{?⃗ =6,8}\)
- \(\color{blue}{?⃗ =4i-3k+2j-5l}\)
- \(\color{blue}{?⃗ =4,-3}\)
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- \(\color{blue}{∥ ?⃗∥}\)\(\color{blue}{=5\sqrt{2}}\)
- \(\color{blue}{∥ ?⃗∥}\)\(\color{blue}{=10}\)
- \(\color{blue}{∥ ?⃗∥}\)\(\color{blue}{=3\sqrt{6}}\)
- \(\color{blue}{∥ ?⃗∥}\)\(\color{blue}{=5}\)
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