Geometric Vectors and Vector Operations

• Geometric vectors are directed line segments characterized by an initial point and a terminal point.
• Magnitude refers to the length of the vector, denoted as $|v|$ or $|v|$.
• The direction of a geometric vector is a key attribute.
• Equality between vectors is defined by having the same magnitude and direction.
• Parallel vectors possess the same or opposite direction.
• The zero vector, represented as $\vec{0}$, has a magnitude of zero and no specific direction.

Vector Operations

• Scalar multiplication, denoted as $\alpha \vec{v}$, scales a vector by a scalar $\alpha$.
  • If $\alpha > 0$, the direction remains the same.
  • If $\alpha < 0$, the direction is reversed.
  • If $\alpha = 0$, the result is the zero vector.
• Vector addition is performed using either the triangle law or the parallelogram law.

Properties of Vector Operations

• Commutative Property: $\vec{u} + \vec{v} = \vec{v} + \vec{u}$
• Associative Property: $\vec{u} + (\vec{v} + \vec{w}) = (\vec{u} + \vec{v}) + \vec{w}$
• Additive Identity: $\vec{u} + \vec{0} = \vec{u}$
• Additive Inverse: $\vec{v} + (-\vec{v}) = \vec{0}$, where $-\vec{v}$ has the same magnitude as $\vec{v}$ but the opposite direction.

River Boat Example

• A river flows due south at 8 km/h, and a motorboat heads east at 12 km/h in still water.
• If $\vec{v}$ = boat velocity and $\vec{w}$ = water velocity, then $\vec{v} + \vec{w}$ is the resultant velocity.
• The magnitude of the resultant velocity is $|\vec{v} + \vec{w}| = \sqrt{8^2 + 12^2} = \sqrt{208} \approx 14.42 \mathrm{~km} / \mathrm{h}$.
• The direction $\theta$ can be found using $\tan \theta = \frac{8}{12} = \frac{2}{3} \Rightarrow \theta = \tan^{-1} (2/3) \approx 33.7^{\circ}$.
• Thus, the boat travels at approximately 14.42 km/h in a direction 33.7° south of east.