Dot product

A team of engineers is designing the support structure for a telecommunications antenna. They have defined two stay cables represented by the vectors $\vec{u} = (2, -3, 1)$ and $\vec{v} = (4, 2, 5)$. Calculate the scalar product of both vectors to study the angle they form at the central node.

Two surveillance drones are in fixed positions in space. The first drone is at point $A(1, 2, -1)$ and the second at point $B(3, -2, 2)$. If the control centre is at the origin of coordinates $O(0, 0, 0)$, calculate the magnitude of the position vector of each drone, $|\vec{OA}|$ and $|\vec{OB}|$.

At a theme park ride, two cables pull a cabin. The directions of the cables are given by the vectors $\vec{u} = (1, 0, 1)$ and $\vec{v} = (0, 1, 1)$. Using the definition of scalar product, find the angle $\alpha$ between the two cables.

Determine the value of the real parameter $k$ such that vectors $\vec{a} = (k, 4, -2)$ and $\vec{b} = (3, -2, 5)$ are perpendicular (orthogonal). This type of calculation is fundamental in video game programming for detecting collisions between surfaces.

Consider the points $A(1, 1, 1)$, $B(3, 1, 1)$ and $C(1, 4, 1)$.

1. a) Find the vectors $\vec{AB}$ and $\vec{AC}$.
2. b) Calculate their scalar product $\vec{AB} \cdot \vec{AC}$.
3. c) Justify whether triangle $ABC$ has a right angle at vertex $A$.

A wind vector has components $\vec{w} = (3, -4, 12)$. In meteorology, it is useful to work with normalised directions. Calculate the unit vector $\vec{u_w}$ in the same direction as $\vec{w}$.

Given the vector $\vec{a} = (6, 2, -3)$, calculate its scalar projection onto the x-axis ($X$-axis). To do this, use the scalar product of $\vec{a}$ with the unit vector of that axis, $\vec{i} = (1, 0, 0)$. Represent the idea of the projection graphically.

You can do it on paper and send a photo with your answer.

In physics, the work done by a constant force is defined as the scalar product of the force vector and the displacement vector: $W = \vec{F} \cdot \vec{d}$. If a force of $\vec{F} = (5, 3, 2)$ newtons displaces an object from point $P(0, 0, 0)$ to point $Q(2, 4, 1)$ metres, calculate the work done expressed in joules.

Let $\vec{u}$ and $\vec{v}$ be two vectors in space such that their magnitudes are $|\vec{u}| = 4$ and $|\vec{v}| = 3$. Given that the angle between them is $60^\circ$, calculate the value of the scalar product of vectors $\vec{a} = 2\vec{u} - \vec{v}$ and $\vec{b} = \vec{u} + 3\vec{v}$ using the distributive and commutative properties.