13.05 Dot product and angle between 2 vectors
Find \mathbf{a} \bullet \mathbf{b} if:
\left|\mathbf{a}\right| = 4, \left|\mathbf{b}\right| = 2 and the angle between \mathbf{a} and \mathbf{b} is 45 \degree.
\left|\mathbf{a}\right| = 2, \left|\mathbf{b}\right| = 5 and the angle between \mathbf{a} and \mathbf{b} is 225 \degree.
\left|\mathbf{a}\right| = 5, \left|\mathbf{b}\right| = 2 and the angle between \mathbf{a} and \mathbf{b} is \dfrac{\pi}{4}.
\left|\mathbf{a}\right| = 4, \left|\mathbf{b}\right| = 5 and the angle between \mathbf{a} and \mathbf{b} is \dfrac{5 \pi}{4}.
Find the dot product of \mathbf{u}=\left(6, 5\right) and \mathbf{v}=\left(7, - 2 \right).
Find \theta, the angle between \mathbf{a} = 12 \mathbf{i} + 9 \mathbf{j} and \mathbf{b} = 15 \mathbf{i} + 8 \mathbf{j}, to the nearest degree.
Show that the vectors \left(3, 2\right) and \left( - 5 , 4\right) are not perpendicular.
\overrightarrow{AB} and \overrightarrow{AC} are perpendicular and are the same length. If \overrightarrow{AB} = \left(1, 4\right), Find \overrightarrow{AC}.
Find the component of \mathbf{a} in the direction of \mathbf{b} if \left|\mathbf{a}\right| = 6, \left|\mathbf{b}\right| = 4 and the angle between \mathbf{a} and \mathbf{b} is 30 \degree.
Find the value(s) of n such that the component of \left(n, 4\right) in the direction of \left(2, 5\right) is:
Equal to 0
Negative
Suppose \mathbf{u} and \mathbf{v} are perpendicular vectors such that \left|\mathbf{u}\right| = 3 \left|\mathbf{v}\right|. Find the (smaller) angle between \mathbf{u} + \mathbf{v} and \mathbf{u} - \mathbf{v}. Round your answer to the nearest degree.
A 22 \text{ N} force in the direction of the vector \left(1, 1\right) moves an object from \left(2, - 1 \right) to \left(6, 5\right).
Write the force vector in terms of \mathbf{i} and \mathbf{j}.
Write the displacement vector in terms of \mathbf{i} and \mathbf{j}.
Find the work done.
Calculate the work done in pushing a table 2.8 \text{ m} across the room against a frictional force of 190 \text{ N}.
Calculate the work done by a person dragging a bin for 90 \text{ m} with a force of 60 \text{ N} at an angle of 12 \degree to the ground. Round your answer to the nearest whole number.
Calculate the work done by gravity in causing a 11 \text{ kg} tree branch to slide 45 \text{ m} down a hill that is at an angle of 43 \degree to the horizontal. Assume the acceleration due to gravity is 9.8 \text{ m/s}^2 . Round your answer to the nearest joule.
A passenger pushes her luggage on the faulty conveyor belt at the airport from \left( - 3 , 0\right) to \left(3, 0\right) (units in metres) with a force of 70 \text{ N} at a 45 \degree angle to the belt. How much work did she do?
A chair is dragged 3 \text{ m} across a room at a 21 \degree angle to the ground. If the amount of work is 126 \text{ J} , find y, the magnitude of the force. Round your answer to the nearest Newton.
Tom wants to push a 60\text{ kg} boulder up a ramp that is inclined at an angle of 17 \degree to the horizontal. If he is to push the boulder at an angle of 36 \degree to the horizontal, what is the minimum force that he must exert? Round your answer to the nearest tenth of a Newton.
Erin pushed a 25 \text{ kg} shelf 5 \text{ m} up a ramp that is inclined at 12 \degree to the horizontal. If she exerted a 80 \text{ N} force at a 28 \degree angle to the ramp, calculate her work done to the nearest joule.
A cafe sold 125 cups of coffee and 145 cups of tea on a particular day. Each cup of coffee was sold for \$3 and each cup of tea was sold for \$2.
Express the number of cups sold as a vector.
Express the prices of the drinks as a vector.
Find the dot product.
What does this dot product represent in context?
Elizabeth is collecting snow samples in a collection box for a science project. Let \mathbf{P} = \mathbf{i} - 3 \mathbf{j} represent the amount of snowfall in centimetres and the direction in which it falls.
Let \mathbf{Q} = 3 \mathbf{i} + 4 \mathbf{j} represent the area in square centimetres and orientation of the opening of the collection box. The total volume of snow collected in the tank is given by V=|\mathbf{P} \bullet \mathbf{Q}|.
Calculate \left|\mathbf{P}\right|, correct to the nearest tenth.
Calculate \left|\mathbf{Q}\right|.
Calculate V.
Describe what your answers to parts (a), (b) and (c) mean in context.
Elizabeth decides to change the orientation of the collection box to maximise the amount of snow collected. Describe how the vectors \mathbf{P} and \mathbf{Q} should be positioned to maximise the amount of snow collected.