Q.For $\vec A = 5\hat i + 12\hat j$ and $\vec B = 3\hat i - 4\hat j$, find

The scalar (dot) product combines two vectors to give an ordinary number: $\vec A\cdot\vec B=AB\cos\theta$, where $\theta$ is the angle between them. It is positive for an acute angle, negative for an obtuse angle, zero for perpendicular vectors, commutative ($\vec A\cdot\vec B=\vec B\cdot\vec A$), and distributive over addition. The self-dot-product gives the magnitude squared, $\vec A\cdot\vec A=A^2$. For the orthogonal unit vectors, $\hat i\cdot\hat i=\hat j\cdot\hat j=\hat k\cdot\hat k=1$ and $\hat i\cdot\hat j=\hat j\cdot\hat k=\hat k\cdot\hat i=0$, which gives the component formula $\vec A\cdot\vec B=A_xB_x+A_yB_y+A_zB_z$. Physically, work done by a constant force through a displacement is the dot product $W=\vec F\cdot\vec d=Fd\cos\theta$.