If \vec{a} = 3 and \vec{b} = 2, and the angle between these two vectors is 60^o, determine \vec{a} \cdot \vec{b}.
A mass of 15 kg is suspended by two cords from a ceiling. The cords have lengths of 15 cm and 20 cm, and the distance between the points where they are attached on the ceiling is 25 cm. Determine the tension in each of the two cords.
In a square that has side lengths of 10 cm, what is the dot product of the vectors representing the diagonals?
An airplane is travelling at 500 km/h due south when it encounters a wind from W45^oN at 100 km/h.
a. What is the resultant velocity of the airplane?
b. How long will it take for the airplane to travel 1000 km?
A 15 kg block lies on a smooth ramp that is inclined at 40° to the ground.
a. Determine the force that this block exerts in a direction perpendicular to the ramp.
b. Whatistheforce,parallel to the inclined plane,needed to prevent the block from slipping?
A regular hexagon, with sides of 3 cm, is shown below. Determine \vec{a}\cdot \vec{ b}.
Given the vectors \vec{a} = (4, -5, 20) and \vec{b} = (1, 2, 2), determine the following:
a. \vec{a} \cdot \vec{b}
b. the cosine of the angle between the two vectors
Given the vectors \vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}, and \vec{c} = 3\vec{i} - \vec{j} - \vec{k}, determine the following:
\vec{a} \cdot \vec{b}
Given the vectors \vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}, and \vec{c} = 3\vec{i} - \vec{j} - \vec{k}, determine the following:
\vec{b} \cdot \vec{c}
Given the vectors \vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}, and \vec{c} = 3\vec{i} - \vec{j} - \vec{k}, determine the following:
\vec{b} + \vec{c}
Given the vectors \vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}, and \vec{c} = 3\vec{i} - \vec{j} - \vec{k}, determine the following:
\vec{a}(\vec{b} + \vec{c})
Given the vectors \vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}, and \vec{c} = 3\vec{i} - \vec{j} - \vec{k}, determine the following:
(\vec{a} + \vec{b})\cdot (\vec{b} + \vec{c})
Given the vectors \vec{a} = \vec{i} + 2\vec{j} + \vec{k}, \vec{b} = 2\vec{i} - 3\vec{j} + 4\vec{k}, and \vec{c} = 3\vec{i} - \vec{j} - \vec{k}, determine the following:
(2\vec{a} - 3\vec{b})\cdot (2\vec{a} + \vec{c})
Given the vectors \vec{p} = x\vec{i} + \vec{j} + 3\vec{k} and \vec{q} = 3x\vec{i} + 10x\vec{j} + \vec{k}, determine the following:
a. the value(s) of x that make these vectors perpendicular
b. the value(s) of x that make these vectors parallel.