Introduction to Dot Product (Scalar Product) Practice Problems

 The scalar product is useful in finding the angle between two vectors. Find the scalar product of vectors M and N for M = 3.00 i + 2.00 j and N = 9.00 i − 4.00 j.

The scalar product is useful in finding the angle between two vectors. Find the angle between the two vectors M and N given M = 5.00 i + 4.00 j and N = 3.00 i − 7.00 j.

Determine the angle θ formed between the given vectors A and B.

A lion stalks a deer grazing in front of a pine tree. When close enough, it attempts a short charge. To reach a safe place, the deer can take two routes. It can go 400 paces north and then 200 paces west, but there are a lot of hyenas on the way, or it can set off along a brick path at a 50° angle west of north. After running 300 paces along the brick path, the deer see a way to the safe place. How far, and in which direction (as an angle east of north) should the deer run to reach the safe place?

A car is moving on a three-dimensional coordinate system, and its velocity is represented by the vector $\overrightarrow{V}$ = (25.0î + 10.0ĵ - 15.0k̂) m/s. Determine the angles that this velocity vector makes with the x, y, and z axes.

Consider three vectors $\overline{P}$ = 3.5ĵ + 2.6k̂, $\overline{Q}$ = 7.2ĵ + 5.1k̂ and $\overline{R}$ lying in the yz plane. Determine $\overline{R}$ such that it has dot products of 12.4 and 14.6 with $\overline{P}$, $\overline{Q}$ respectively.

Vector $\overline{A}$ having a magnitude of 60 lies along the positive y-axis. Vector $\overline{B}$, positioned in the xy-plane, forms a 67° angle above the positive x-axis and has a magnitude of 42. Determine the dot product $\overline{A}\cdot\overline{B}$.

Vector $\overline{M}$ lies in the xy plane making an angle of θ above the positive x-axis. Vector $\overline{N}$ is lying in the same plane making an angle Φ below the positive x-axis. Determine their dot product.

Hint: Use one of the following identities-

cos(A + B) = cosAcosB - sinAsinB

cos(A - B) = cosAcosB + sinAsinB

In an orchard, the trees are planted in a hexagonal pattern for optimal spacing. Tree 1 is located at coordinates (0, 0, 0), Tree 2 at (8.4 m, 5.6 m, 0), and Tree 3 at (3.7 m, 7.9 m, 2.3 m). Calculate the angle between two vectors: one connecting Tree 1 to Tree 2, and the other linking Tree 1 to Tree 3.