Understand and express a vector in the plane in polar form using the notation (𝑟, 𝜃).

Determine the polar form $(r, \theta)$ of the vector $\vec{v}$ shown in the diagram, where $\theta$ is measured anticlockwise from the positive $x$-axis and $0 \leq \theta < 2\pi$.

A helicopter flies from point A to point B. Its displacement vector in Cartesian form is shown in the diagram below, where the horizontal axis represents east and the vertical axis represents north. a) Determine the magnitude of the displacement vector. (1 mark) b) Determine the direction angle θ measured anticlockwise from the positive x-axis (east direction). Express your answer in degrees, correct to 1 decimal place. (1 mark) c) Hence express the displacement vector in polar form using the notation (r, θ). (1 mark)

A vector $\vec{v}$ has Cartesian components $\vec{v} = -6\vec{i} + 8\vec{j}$. a) Determine the magnitude $r$ of this vector. (1 mark) b) Determine the direction angle $\theta$ of this vector, measured anticlockwise from the positive $x$-axis, where $0° \leq \theta < 360°$. (1 mark) c) Express the vector $\vec{v}$ in polar form using the notation $(r, \theta)$. (1 mark)

A surveying team measures displacement vectors from a base station to three control points. The table in the stimulus shows the Cartesian components of each vector. (a) Determine the polar form (r, θ) of vector v₁, where 0 ≤ θ < 2π. Express r correct to three decimal places and θ in radians correct to three decimal places. (b) Determine the polar form (r, θ) of vector v₂, where 0 ≤ θ < 2π. Express r correct to three decimal places and θ in radians correct to three decimal places. (c) Show that the angle between vectors v₁ and v₃ is π/4 radians.

A displacement vector \(\vec{AB}\) has Cartesian components \(\vec{AB} = 6\mathbf{i} - 8\mathbf{j}\). a) Determine the magnitude \(r\) of this vector. (1 mark) b) Determine the direction angle \(\theta\) of this vector, measured anticlockwise from the positive \(x\)-axis, where \(0^\circ \leq \theta < 360^\circ\). (1 mark) c) Express the vector \(\vec{AB}\) in polar form using the notation \((r, \theta)\). (1 mark)