Vectors. Definitions
A force, a velocity, a displacement... it is not enough to say "how much": we also need to say "which way". That is what distinguishes a vector quantity from a scalar one, and the mathematical object that captures it is the vector.
Scalar and vector quantities
Classify the following quantities used in physics and geometry as scalar or vector quantities, briefly justifying your answer based on their definitions: Temperature of…
Classify the following quantities used in physics and geometry as scalar or vector quantities, briefly justifying your answer based on their definitions:
-
Temperature of a data server ($45^\circ \text{C}$).
-
Wind velocity at a wind farm ($20 \text{ km/h}$ in a North-West direction).
-
Mass of an artificial satellite ($1200 \text{ kg}$).
-
Displacement of a drone from its base to a delivery point.
The fixed vector $\vec{AB}$
A robotic arm on an assembly line moves in a two-dimensional Cartesian plane. If its initial position is the point $A(3, -2)$ and its final position after a movement is…
A robotic arm on an assembly line moves in a two-dimensional Cartesian plane. If its initial position is the point $A(3, -2)$ and its final position after a movement is $B(-1, 5)$, determine the components of the displacement vector $\vec{AB}$.
A fishing boat is located at coordinates $P(-4, 2)$ on a nautical chart. If the captain programmes the autopilot to make a straight-line movement defined by the vector…
A fishing boat is located at coordinates $P(-4, 2)$ on a nautical chart. If the captain programmes the autopilot to make a straight-line movement defined by the vector $\vec{v} = (8, -3)$, determine the coordinates of the destination point $Q$.
Magnitude, direction and sense
In a video game design, the velocity of a character is given by the vector $\vec{v} = (-6, 8)$ in pixels per frame. Calculate the magnitude of this vector to determine…
In a video game design, the velocity of a character is given by the vector $\vec{v} = (-6, 8)$ in pixels per frame. Calculate the magnitude of this vector to determine the character's actual speed on the screen.
To install a fibre optic cable between two buildings, surveyors mark the positions of the connections on a plan as $A(-3, 5)$ and $B(2, -7)$, with units in metres.…
To install a fibre optic cable between two buildings, surveyors mark the positions of the connections on a plan as $A(-3, 5)$ and $B(2, -7)$, with units in metres. Calculate the straight-line distance between both points.
Equipollent vectors
Observe the vectors represented in the following grid. Identify which pairs of vectors are equipollent (equal), which have the same magnitude but different direction,…
Observe the vectors represented in the following grid. Identify which pairs of vectors are equipollent (equal), which have the same magnitude but different direction, and which have the same direction but opposite sense.
The position vector
A support pillar is to be placed exactly at the centre of a steel beam. If the ends of the beam are located at the coordinates $M(-1, 4)$ and $N(7, -2)$, determine the…
A support pillar is to be placed exactly at the centre of a steel beam. If the ends of the beam are located at the coordinates $M(-1, 4)$ and $N(7, -2)$, determine the coordinates of the point where the pillar should go.
Three consecutive vertices of an urban garden are located at coordinates $A(1, 2)$, $B(5, 3)$ and $C(6, 7)$, expressed in decametres. The garden is intended to be in the…
Three consecutive vertices of an urban garden are located at coordinates $A(1, 2)$, $B(5, 3)$ and $C(6, 7)$, expressed in decametres. The garden is intended to be in the shape of a parallelogram $ABCD$. Determine the coordinates where the fourth vertex $D$ should be located.