Dot Product Exercises

Calculate the scalar product of the vectors $\vec{u} = (3, -2)$ and $\vec{v} = (-1, 5)$.

Knowing that the magnitudes of two vectors are $|\vec{a}| = 4$ and $|\vec{b}| = 6$, and that the angle between them is $60^\circ$, determine their scalar product.

Find the magnitude of the vector $\vec{w} = (-5, 12)$.

Determine, justifying your answer by means of a mathematical calculation, whether the vectors $\vec{p} = (4, 6)$ and $\vec{q} = (-3, 2)$ are orthogonal.

Calculate the value of the parameter $k$ so that the vectors $\vec{u} = (k, 3)$ and $\vec{v} = (2, -4)$ are perpendicular.

Find the coordinates of a vector that is perpendicular to $\vec{v} = (-7, 2)$ and has exactly the same magnitude.

Calculate the angle between the vectors $\vec{a} = (1, \sqrt{3})$ and $\vec{b} = (-\sqrt{3}, 1)$.

To stabilise the mast of a telecommunications antenna, two cables are tensioned from the same point on the mast and anchored to the ground. The vectors representing the direction of the cables from the top anchor to the ground are $\vec{T_1} = (-4, -5)$ and $\vec{T_2} = (6, -3)$. Calculate the angle between the two cables at the top anchor point.

Determine the possible values of the parameter $m$ so that the vectors $\vec{u} = (m, 1)$ and $\vec{v} = (2, m)$ form an angle of $30^\circ$.

In the physics engine of a video game, a character located at coordinate $P(10, 15)$ must fire a projectile in the direction of vector $\vec{d} = (3, -4)$ at a constant speed of $20$ distance units per frame. Determine the exact velocity vector $\vec{v}$ of the projectile, knowing that it must have the same direction and sense as $\vec{d}$ but with magnitude $20$.

Find the coordinates of a vector $\vec{w} = (x, y)$ knowing that its magnitude is $\sqrt{13}$ and that its scalar product with the vector $\vec{v} = (2, 3)$ gives $13$.

Find the coordinates of all vectors $\vec{a} = (x, y)$ that form an exact angle of $45^\circ$ with the vector $\vec{b} = (2, -2)$ and whose magnitude is exactly half the magnitude of $\vec{b}$.

In physics, the work $W$ done by a constant force $\vec{F}$ in moving an object along a displacement vector $\vec{d}$ is calculated using the scalar product $W = \vec{F} \cdot \vec{d}$. If a crane applies a force of $\vec{F} = (150, -40)$ newtons to move a container from point $A(2, 5)$ to point $B(12, 5)$ (coordinates expressed in meters), calculate the total work done.

Calculate the orthogonal projection vector of the vector $\vec{u} = (4, -3)$ onto the vector $\vec{v} = (1, 2)$.

In the installation of a solar panel, the normal vector (perpendicular to the surface of the panel) is given by $\vec{n} = (k, \sqrt{3})$. Sun rays strike the panel along the vector $\vec{s} = (1, -1)$. Calculate the positive value of the parameter $k$ so that the angle of incidence (the angle formed between the normal vector $\vec{n}$ and the vector opposite to the sun rays $-\vec{s}$) is $60^\circ$.

Calculate the distance between the points $A(2, -1)$ and $B(5, 3)$ by first determining the components of the vector $\vec{AB}$.

Check, using only the scalar product, whether the triangle defined by the vertices $A(1, 2)$, $B(4, 3)$ and $C(3, -1)$ is a right-angled triangle. If so, state at which vertex the right angle is located.

Given the points $A(-2, 1)$ and $B(4, 5)$, find the coordinates of a point $C$ situated on the x-axis such that the triangle $ABC$ is right-angled with the right angle at vertex $C$. How many possible solutions are there for point $C$?

Given the vectors $\vec{a} = (3, -1)$ and $\vec{b} = (1, 3)$, express the vector $\vec{c} = (7, 1)$ as a linear combination of $\vec{a}$ and $\vec{b}$. Once solved, calculate the scalar product of $\vec{a}$ and $\vec{b}$ and interpret the result geometrically.

For the vectors $\vec{u}$ and $\vec{v}$, we know that $|\vec{u}| = 3$, $|\vec{v}| = 5$ and that they form an angle of $120^\circ$. Applying the properties of the scalar product, calculate the exact value of the expression $(2\vec{u} - \vec{v}) \cdot (\vec{u} + 3\vec{v})$.

Knowing that $\vec{u}$ and $\vec{v}$ are two unit vectors, determine the angle they must form with each other so that the sum vector $\vec{u} + \vec{v}$ has a magnitude equal to $\sqrt{3}$.

An industrial robotic arm moves in a two-dimensional plane. Its gripper is initially at the point $P(2, 5)$ and must move in a straight line to a point $Q$ located on a conveyor belt whose path is defined by the line $x - 2y + 4 = 0$. If the displacement vector $\vec{PQ}$ must be orthogonal to the direction of the conveyor belt, determine the exact coordinates of the point $Q$.

Prove vectorially, using the properties of the scalar product, that the diagonals of any rhombus intersect perpendicularly. To do so, consider a rhombus generated by two free vectors $\vec{u}$ and $\vec{v}$ starting from the same vertex and satisfying $|\vec{u}| = |\vec{v}|$.

Calculate the exact area of the triangle determined by the vertices $A(1, 1)$, $B(5, 2)$ and $C(3, 6)$. To do so, use the orthogonal projection of the vector $\vec{AC}$ onto the vector $\vec{AB}$ to determine the height of the triangle, and the magnitude of $\vec{AB}$ as the base.