Exercises: Scalar product exercises

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

Given that the moduli 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 modulus of the vector $\vec{w} = (-5, 12)$.

Determine, justifying your answer with a mathematical operation, 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.

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

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

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

In physics, the work $W$ done by a constant force $\vec{F}$ when displacing 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 feet), calculate the total work done.

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

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 a video game physics engine, a character located at the coordinate $P(10, 15)$ must fire a projectile in the direction of the 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 a modulus of $20$.

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

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

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.

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

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 $(2\vec{u} - \vec{v}) \cdot (\vec{u} + 3\vec{v})$.

An industrial robotic arm moves in a two-dimensional plane. Its gripper is initially located 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 this, consider a rhombus generated by two free vectors $\vec{u}$ and $\vec{v}$ that start from the same vertex and satisfy $|\vec{u}| = |\vec{v}|$.

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

In the installation of a solar panel, the normal vector (perpendicular to the surface of the panel) is given by $\vec{n} = (k, \sqrt{3})$. The sun's rays strike the panel according to 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 opposite vector to the sun's rays $-\vec{s}$) is $60^\circ$.

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

Given 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 modulus equal to $\sqrt{3}$.

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