Q.Three vectors $\vec a$, $\vec b$ and $\vec c$ satisfy the condition $\vec a + \vec b + \vec c = \vec 0$. If $|\vec a| = 1$, $|\vec b| = 4$ and $|\vec c| = 3$, then evaluate $\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a$.

Vector Magnitude Properties

An arrow has a direction and a length. That length — the straight-line distance from tail to tip — is the magnitude of the vector, written $|\vec{v}|$ or $\|\vec{v}\|$. It is always non-negative and tells you how much of something there is, ignoring direction.

Definition

Magnitude is the distance from the origin to the point the vector reaches — the Pythagorean theorem in $n$ dimensions:

The four key properties

1. Non-negativity.

A length is never negative, and only the zero vector has zero length.

2. Scaling.

Stretching a vector by $k$ multiplies its length by $|k|$ — the absolute value appears because a negative $k$ flips direction but the length still grows by $|k|$. E.g. if $|\vec{v}| = 3$, then $|-2\vec{v}| = 2\times 3 = 6$.

3. Triangle inequality.

The direct path is never longer than going the long way: the straight line from $A$ to $C$ is at most the distance $A \to B \to C$. Equality holds only when $\vec{u}$ and $\vec{v}$ point in exactly the same direction.

4. Dot-product relation.

The squared length equals the vector's dot product with itself, since $\vec{v}\cdot\vec{v} = x^2 + y^2 + z^2$. This is the workhorse in proofs and in physics (kinetic energy $\tfrac{1}{2}m|\vec{v}|^2$). …