Dot Product of Vectors in Space

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 center 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.

The vertices of a triangle in space are $A(1, 2, 1)$, $B(3, -1, 0)$ and $C(-1, 0, 4)$.

• a) Calculate the interior angle at vertex $A$.

• b) Use the scalar product to determine whether the triangle is acute-angled, right-angled or obtuse-angled.

Determine the value of $k$ so that the angle formed by vectors $\vec{u} = (1, 1, 0)$ and $\vec{v} = (0, 1, k)$ is exactly $45^\circ$. Justify whether the obtained value of $k$ is unique or if there are several possible orientations for vector $\vec{v}$.

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 $A$.

Given the vectors $\vec{u} = (m, 2, -1)$ and $\vec{v} = (3, m, m+2)$, determine the values of the real parameter $m$ such that both vectors are orthogonal. If several values exist, explain whether the resulting vectors in each case have the same magnitude.

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 $Q(2, 4, 1)$ meters, calculate the work done expressed in joules.

Calculate the scalar projection and the vector projection of the vector $\vec{a} = (1, -2, 5)$ onto the vector $\vec{b} = (3, 0, 4)$.

1. a) Calculate the scalar projection of $\vec{a}$ onto $\vec{b}$.

1. b) Calculate the vector projection of $\vec{a}$ onto $\vec{b}$.

1. c) Interpret graphically whether the projection vector has the same direction as $\vec{b}$, based on the sign of the dot product $\vec{a}\cdot\vec{b}$ that you obtained.

A vector $\vec{w}$ has magnitude 10 and its first two direction cosines are $\cos \alpha = 0.6$ and $\cos \beta = -0.8$.

• a) Determine the third direction cosine, $\cos \gamma$.

• b) Find the components of the vector $\vec{w}$.

In an engineering structure, a force of $\vec{F} = (4, -3, 2)$ newtons acts on a point, displacing it from position $P(1, 1, 1)$ to $Q(5, 2, -3)$ (coordinates in meters). Calculate the work done by the force, recalling that work is the scalar product of the force vector and the displacement vector $\vec{W} = \vec{F} \cdot \vec{d}$.

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.

Let $\vec{u}$ and $\vec{v}$ be two unit vectors. Prove that the sum vector $\vec{u} + \vec{v}$ and the difference vector $\vec{u} - \vec{v}$ are always perpendicular to each other, regardless of the components of these vectors. Use this result to explain the geometrical relationship between the diagonals of a rhombus.

Prove, using the properties of the scalar product, that for any two vectors $\vec{x}, \vec{y} \in \mathbb{R}^3$, the following identity (known as the Polarisation Identity) holds:

Next, use this result to calculate the scalar product of two vectors whose sum and difference have magnitudes $|\vec{x} + \vec{y}| = 10$ and $|\vec{x} - \vec{y}| = 6$. What angle do these vectors form if we also know that their magnitudes are equal?

Let $B = \{ \vec{e}_1, \vec{e}_2, \vec{e}_3 \}$ be a basis of $\mathbb{R}^3$ that is not necessarily orthonormal. The following scalar products between the basis vectors are known:

• $\vec{e}_1 \cdot \vec{e}_1 = 2$, $\vec{e}_2 \cdot \vec{e}_2 = 1$, $\vec{e}_3 \cdot \vec{e}_3 = 3$

• $\vec{e}_1 \cdot \vec{e}_2 = -1$, $\vec{e}_1 \cdot \vec{e}_3 = 0$, $\vec{e}_2 \cdot \vec{e}_3 = 1$

Calculate the magnitude of vector $\vec{u} = 2\vec{e}_1 - 3\vec{e}_2 + \vec{e}_3$ and determine the cosine of the angle that this vector $\vec{u}$ forms with vector $\vec{e}_1$.

Given the vectors $\vec{u} = (m, 1, 1)$ and $\vec{v} = (1, m, 1)$:

1. a) Determine the values of the real parameter $m$ for which the angle formed by $\vec{u}$ and $\vec{v}$ is exactly $60^\circ$.

1. b) For the value of $m=0$, find a vector $\vec{w}$ with magnitude $\sqrt{6}$ that is a linear combination of $\vec{u}$ and $\vec{v}$, and is also orthogonal to vector $\vec{a} = (1, 1, -1)$.

An artificial satellite moves along a straight trajectory. At a given instant, its position vector is $\vec{r} = (2, 2, 1)$ and its velocity vector is $\vec{v} = (1, -2, 2)$. A control center on Earth emits a radio signal whose direction is given by vector $\vec{s} = (k, 1, 0)$.

Calculate the value of parameter $k$ so that the projection of the satellite's velocity onto the direction of the radio signal is zero. Interpret physically what it means for this projection to be zero in terms of the satellite approaching or moving away from the signal source.

In the design of a state-of-the-art architectural structure, a tension cable is required to connect a fixed point $P(1, 2, 3)$ to a variable point $Q$ located on a metal beam whose path is given by line $r: (x, y, z) = (2, 0, 1) + \lambda(1, -1, 0)$.

It is desired that the cable be as short as possible. Use the concepts of scalar product and projection to:

1. a) Determine the coordinates of point $Q$ that minimises the length of the cable.
2. b) Check that, for this point, vector $\vec{PQ}$ is perpendicular to the direction vector of beam $r$.

Consider the points in space $A(1, 0, -1)$ and $B(3, 2, 1)$. Let $r$ be the line passing through the point $C(0, 4, 3)$ with direction vector $\vec{d} = (1, 1, 1)$.

Determine whether there exists any point $P$ belonging to the line $r$ such that the vectors $\vec{AP}$ and $\vec{BP}$ are orthogonal. Is there a unique solution? Justify your answer by analysing the resulting quadratic equation.

A team of engineers is calibrating the orientation sensor of an industrial inspection drone. The reference system is fixed at the take-off base. It is known that the main axis of the drone must follow the direction of a vector $\vec{w}$ that simultaneously satisfies the following three conditions:

1. Vector $\vec{w}$ must be a unit vector.

2. $\vec{w}$ must be orthogonal to vector $\vec{u} = (1, -2, 2)$, which represents the direction of the prevailing wind.

3. The angle formed by $\vec{w}$ with the position vector of the target, $\vec{v} = (0, 3, 4)$, must be $45^\circ$.

Determine the coordinates of the possible vectors $\vec{w}$ that the drone's navigation software can use.

Consider a tetrahedron with vertices $O(0,0,0)$, $A(a, 0, 0)$, $B(0, b, 0)$ and $C(0, 0, c)$ with $a, b, c > 0$. Let $M$ be the midpoint of the segment $BC$.

1. a) Using the scalar product exclusively, prove that vector $\vec{AM}$ is perpendicular to segment $BC$ if and only if the legs of the base along axes $Y$ and $Z$ have the same length ($b = c$).

1. b) If we fix $a=4$, $b=2$ and $c=2$, calculate the orthogonal projection of vector $\vec{OA}$ onto the direction defined by vector $\vec{AB}$. Express the result as a vector.