Find \(|\vec{a}|\) and \(|\vec{b}|\) ,if \((\vec{a}+\vec{b}).(\vec{a}-\vec{b})=8\) and \(|\vec{a}|=8|\vec{b}|.\)
Solution and Explanation
\((\vec{a}.\vec{b}).(\vec{a}-\vec{b})=8\)
\(⇒\vec{a}.\vec{a}-\vec{a}.\vec{b}+\vec{b}.\vec{a}-\vec{b}.\vec{b}=8\)
\(⇒|\vec{a}|^2-|\vec{b}|^2=8\)
\(⇒(8|\vec{b}|)^2-|\vec{b}|^2=8 [|\vec{a}|=8|\vec{b}|]\)
\(⇒64|\vec{b}|^2-|\vec{b}|^2=8\)
\(⇒63|\vec{b}|^2=8\)
\(⇒|\vec{b}|^2=\frac{8}{63}\)
\(⇒|\vec{b}|^=\sqrt{\frac{8}{63}}\) [Magnitude of a vector is non-negative]
\(⇒|\vec{b}|=\frac{2\sqrt{2}}{3\sqrt{7}}\)
\(⇒|\vec{a}|=8|\vec{b}|=\frac{8×2\sqrt{2}}{3\sqrt{7}}=\frac{16\sqrt{2}}{3\sqrt{7}}.\)
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Concepts Used:
Product of Two Vectors
A vector is an object that has both the direction and the magnitude. The length indicates the magnitude of the vectors, whereas the arrow indicates the direction. There are different types of vectors such as:
- Dot product of vectors (Scalar product)
- Cross product of vectors (Vector product)
A vector product is a cross-product or area product, which is formed when two real vectors are joined together in a three-dimensional space. If we assume the two vectors to be a and b, their vector is denoted by a x b.
The Magnitude of the Vector Product:
|c¯| = |a||b|sin θ
Where;
a and b are the magnitudes of the vector and θ is equal to the angle between the two given vectors. In this way, we can say that there are two angles between any two given vectors.
These two angles are θ and (360° - θ). When we follow this rule we consider the smaller angle which is less than 180°.