Parallel vectors and applications

Velocity and bearings, parallel and collinear vectors, linear combinations and linear dependence. 8 problems with step-by-step solutions.

Solve each problem in your notebook first, then open the solution.

Problem 1 Bearings and velocity

a) A ship sails in the direction N 30° E at 40 km/h. Find the east and north components of its velocity.

b) An airplane heads due north at 300 km/h and the wind blows from the west at 40 km/h. Find the speed of the plane relative to the ground and its direction.

Show solution

a) East: \(40\sin 30^\circ = 20\) km/h; north: \(40\cos 30^\circ = 20\sqrt3 \approx 34.6\) km/h.

b) Wind from the west blows to the east. Ground velocity \((40, 300)\): speed \(\sqrt{1600 + 90000} \approx 302.7\) km/h, direction N \(\theta\) E with \(\tan\theta = \frac{40}{300}\), \(\theta \approx 7.6^\circ\).

Problem 2 Crossing a river

A boat moves at 10 km/h in still water. The river flows east at 5 km/h. In which direction must the boat head to reach the point directly opposite on the north bank? What is its speed over the ground?

Show solution

The east component of the boat's own velocity must cancel the current: \(10\sin\theta = 5\), \(\theta = 30^\circ\). Head N 30° W.

Ground speed: \(10\cos 30^\circ = 5\sqrt3 \approx 8.66\) km/h, due north.

Problem 3 Parallel vectors

a) Show that \(\vec{u} = (6, -4)\) and \(\vec{v} = (-9, 6)\) are parallel.

b) \(\vec{a} = (m - 2, 3)\) and \(\vec{b} = (4, m + 2)\) are parallel. Find \(m\).

c) \(A(1, -2)\), \(B(3, 2)\), \(C(4, -1)\), \(D(x, y)\) and \(CD \parallel AB\). Find the relation between \(x\) and \(y\).

Show solution

a) \(\vec{v} = -\frac32\vec{u}\), so they are parallel (also \(\frac{6}{-9} = \frac{-4}{6}\)).

b) \(\frac{m - 2}{4} = \frac{3}{m + 2} \Rightarrow m^2 - 4 = 12 \Rightarrow m = \pm 4\).

c) \(\vec{AB} = (2, 4)\), \(\vec{CD} = (x - 4, y + 1)\). Parallel: \(4(x - 4) = 2(y + 1)\), i.e. \(y = 2x - 9\).

Problem 4 Collinear points and midpoints

a) Are \(A(-1, 3)\), \(B(1, 2)\) and \(C(7, -1)\) collinear?

b) A triangle has vertices \(A(4, 6)\), \(B(-2, 2)\), \(C(6, -4)\). \(E\) and \(F\) are the midpoints of \(AB\) and \(AC\). Find \(E\), \(F\) and show that \(\vec{EF} = \frac12\vec{BC}\).

Show solution

a) \(\vec{AB} = (2, -1)\), \(\vec{AC} = (8, -4) = 4\vec{AB}\). They are parallel and share \(A\): collinear.

b) \(E(1, 4)\), \(F(5, 1)\). \(\vec{EF} = (4, -3)\) and \(\vec{BC} = (8, -6)\), so \(\vec{EF} = \frac12\vec{BC}\).

Problem 5 Linear combination

a) Write \(\vec{w} = (7, 1)\) as a linear combination of \(\vec{u} = (1, 2)\) and \(\vec{v} = (3, -1)\).

b) Write \(\vec{x} = (1, 8)\) as a linear combination of \(\vec{a} = (2, -1)\) and \(\vec{b} = (-1, 3)\).

c) Find \(\vec{w} = 2\vec{u} - 3\vec{v}\) for \(\vec{u} = (4, -1)\), \(\vec{v} = (1, 2)\).

Show solution

a) \(\vec{w} = c_1\vec{u} + c_2\vec{v}\): \(c_1 + 3c_2 = 7\), \(2c_1 - c_2 = 1\). So \(c_1 = \frac{10}{7}\), \(c_2 = \frac{13}{7}\). Check: \(\frac{10}{7} + \frac{39}{7} = 7\), \(\frac{20}{7} - \frac{13}{7} = 1\).

b) \(2c_1 - c_2 = 1\), \(-c_1 + 3c_2 = 8\): \(c_1 = \frac{11}{5}\), \(c_2 = \frac{17}{5}\).

c) \((8, -2) - (3, 6) = (5, -8)\).

Problem 6 Linear dependence

a) Show that \(\{(2, 1), (1, 3)\}\) is linearly independent.

b) Show that \(\{(1, 0), (0, 1), (2, -5)\}\) is linearly dependent.

c) \(\vec{a} = (k, 3)\) and \(\vec{b} = (2, k + 1)\) are linearly dependent. Find \(k\).

Show solution

a) \(c_1(2, 1) + c_2(1, 3) = (0, 0)\): \(2c_1 + c_2 = 0\), \(c_1 + 3c_2 = 0\), only \(c_1 = c_2 = 0\). Independent.

b) \(2(1, 0) - 5(0, 1) - (2, -5) = (0, 0)\) with non-zero coefficients. Dependent.

c) Two vectors are dependent when they are parallel: \(k(k + 1) = 6\), so \(k = 2\) or \(k = -3\).

Problem 7 Parallelogram with vectors

\(ABCD\) is a parallelogram, \(\vec{OA} = 2\vec{i} + \vec{j}\), \(\vec{AB} = 4\vec{i} + \vec{j}\) and \(\vec{AD} = \vec{i} + 3\vec{j}\). Find the coordinates of the four vertices and the intersection point of the diagonals.

Show solution

\(A(2, 1)\), \(B = A + \vec{AB} = (6, 2)\), \(D = A + \vec{AD} = (3, 4)\), \(C = A + \vec{AB} + \vec{AD} = (7, 5)\).

The diagonals meet at the midpoint of \(AC\): \(\left(\frac92, 3\right)\).

Problem 8 Mixed problems

a) \(\vec{PQ} = (8, -15)\) and \(Q(3, -6)\). Find \(|\vec{PQ}|\) and the point \(P\).

b) \(\vec{RS}\) is parallel to \(\vec{PQ}\) and \(|\vec{RS}| = 34\). Write the two possible vectors \(\vec{RS}\).

c) \(\vec{A} = (2a + 1)\vec{i} + 4\vec{j}\) and \(\vec{B} = 3\vec{i} + (a + 1)\vec{j}\) are parallel. Find \(a\).

Show solution

a) \(|\vec{PQ}| = \sqrt{64 + 225} = 17\). \(P = Q - \vec{PQ} = (-5, 9)\).

b) \(\vec{RS} = \pm 2\vec{PQ} = (16, -30)\) or \((-16, 30)\).

c) \((2a + 1)(a + 1) = 12 \Rightarrow 2a^2 + 3a - 11 = 0 \Rightarrow a = \frac{-3 \pm \sqrt{97}}{4}\).