Q.If $\vec a$ and $\vec b$ are two vectors such that $|\vec a|=10,|\vec b|=15$ and $\vec a\cdot\vec b=75\sqrt2$, find the angle between $\vec a$ and $\vec b$.

For non-zero vectors $\vec a,\vec b$ with included angle $\theta$ ($0\le\theta\le\pi$), the scalar (dot) product is the number $\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta.$

Geometric meaning (projection). $\vec a\cdot\vec b=|\vec a|\times(\text{projection of }\vec b\text{ on }\vec a)$, and the projection of $\vec b$ on $\vec a$ is $\dfrac{\vec a\cdot\vec b}{|\vec a|}$ (symmetrically, projection of $\vec a$ on $\vec b$ is $\dfrac{\vec a\cdot\vec b}{|\vec b|}$).

Core properties.

• Commutative: $\vec a\cdot\vec b=\vec b\cdot\vec a$.
• Sign follows the angle: positive for $0\le\theta<\pi/2$, zero at $\theta=\pi/2$, negative for $\pi/2<\theta\le\pi$. In particular $\vec a\cdot\vec b=0 \iff \vec a=\vec 0$ or $\vec b=\vec 0$ or $\vec a\perp\vec b$ — for two non-zero vectors, $\vec a\cdot\vec b=0$ is exactly the perpendicularity test.
• $\vec a\cdot\vec a=|\vec a|^2$ (often written $a^2$), so $|\vec a|=\sqrt{\vec a\cdot\vec a}$.
• $\hat i\cdot\hat i=\hat j\cdot\hat j=\hat k\cdot\hat k=1$ and $\hat i\cdot\hat j=\hat j\cdot\hat k=\hat k\cdot\hat i=0$ (they're mutually perpendicular unit vectors).
• Distributive: $\vec a\cdot(\vec b+\vec c)=\vec a\cdot\vec b+\vec a\cdot\vec c$, and likewise for subtraction and for the right factor.
• Identities (proved exactly like $(x+y)^2$ for numbers): $|\vec a+\vec b|^2=|\vec a|^2+|\vec b|^2+2\vec a\cdot\vec b$; $|\vec a-\vec b|^2=|\vec a|^2+|\vec b|^2-2\vec a\cdot\vec b$; $(\vec a+\vec b)\cdot(\vec a-\vec b)=|\vec a|^2-|\vec b|^2$.
• Coordinate formula: for $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$, $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$: $\vec a\cdot\vec b=a_1b_1+a_2b_2+a_3b_3.$
• Angle formula: $\theta=\cos^{-1}\!\left(\dfrac{\vec a\cdot\vec b}{|\vec a||\vec b|}\right)$.
• Triangle/Cauchy–Schwarz-type inequalities: $|\vec a+\vec b|\le|\vec a|+|\vec b|$ and $|\vec a\cdot\vec b|\le|\vec a||\vec b|$.

Because the dot product pins down the angle unambiguously between $0$ and $\pi$, it is the preferred tool whenever a problem asks for 'the angle between two vectors' (the cross product only ever returns the acute angle, since $\sin\theta\ge0$ throughout $[0,\pi]$).