Calculate the magnitude and direction of a vector. |𝒂| = |(𝑎1 𝑎2)| = √𝑎12 + 𝑎22 tan(𝜃) = 𝑦 𝑥, 𝑥 ≠ 0

What is the magnitude of the vector $\mathbf{p} = 3\mathbf{i} - 5\mathbf{j}$?

A yacht sails with velocity **v** = 8**i** + 15**j** m s⁻¹, where **i** and **j** are unit vectors pointing east and north respectively. a) Calculate the speed of the yacht. (1 mark) b) The direction of motion makes an angle θ with the eastward direction, where tan(θ) = (northward component)/(eastward component). Calculate θ in degrees, correct to one decimal place. (2 marks)

Calculate the magnitude of vector $\mathbf{PQ}$.

A yacht navigates from a harbour using a course determined by two consecutive displacement vectors. The first displacement is represented by **p** = 5**i** + 12**j** and the second by **q** = –8**i** + 6**j**, where distances are measured in kilometres. a) Calculate the magnitude of the resultant displacement vector **r** = **p** + **q**. b) Calculate the direction of **r** as a bearing, measured clockwise from north. Express your answer correct to the nearest degree.

A particle moves in a plane such that its position vector at time $t$ seconds is given by $\mathbf{r}(t) = (3t^2 - 4t)\mathbf{i} + (2t^2 + 5t - 1)\mathbf{j}$. (a) Determine the velocity vector $\mathbf{v}$ when $t = 3$. (2 marks) (b) Calculate the magnitude and direction of the velocity vector found in part (a). Express the direction as an angle measured anticlockwise from the positive $x$-axis, correct to one decimal place. (2 marks)