Edexcel IAL - Pure Maths 4- 7.2 Magnitude of a Vector- Study notes - New syllabus
Edexcel IAL – Pure Maths 4- 7.2 Magnitude of a Vector -Study notes- New syllabus
Edexcel IAL – Pure Maths 4- 7.2 Magnitude of a Vector -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 7.2 Magnitude of a Vector
Magnitude of a Vector
The magnitude of a vector is its length.
If a vector is written as:
\( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \)
then its magnitude is:
\( |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \)
The notation \( |\mathbf{a}| \) means the length of vector \( \mathbf{a} \).
Unit Vector in the Direction of \( \mathbf{a} \)
A unit vector has magnitude 1.
The unit vector in the direction of \( \mathbf{a} \) is obtained by dividing the vector by its magnitude:
\( \dfrac{\mathbf{a}}{|\mathbf{a}|} \)
Geometric Meaning
- \( |\mathbf{a}| \) tells how long the vector is.
- \( \dfrac{\mathbf{a}}{|\mathbf{a}|} \) gives the direction of the vector only.
Example
Find the magnitude of \( \mathbf{a} = 3\mathbf{i} + 4\mathbf{j} \).
▶️ Answer / Explanation
\( |\mathbf{a}| = \sqrt{3^2 + 4^2} = 5 \)
Example
Find a unit vector in the direction of \( \mathbf{a} = 6\mathbf{i} – 8\mathbf{j} \).
▶️ Answer / Explanation
\( |\mathbf{a}| = \sqrt{36 + 64} = 10 \)
Unit vector \( = \dfrac{6\mathbf{i}-8\mathbf{j}}{10} = \dfrac{3}{5}\mathbf{i} – \dfrac{4}{5}\mathbf{j} \)
Example
Find the magnitude and unit vector of \( \mathbf{a} = 2\mathbf{i} – 3\mathbf{j} + 6\mathbf{k} \).
▶️ Answer / Explanation
\( |\mathbf{a}| = \sqrt{4+9+36} = 7 \)
Unit vector \( = \dfrac{2\mathbf{i}-3\mathbf{j}+6\mathbf{k}}{7} \)