2.1 Vectors in the plane  (Page 10/29)

Find $x\in ℝ$ such that vectors $\mathbf{\text{a}}=⟨x,\text{sin}x⟩$ and $\mathbf{\text{b}}=⟨x,\text{cos}x⟩$ are equivalent.

Calculate the coordinates of point $D$ such that $ABCD$ is a parallelogram, with $A(1,1),$ $B(2,4),$ and $C(7,4).$

Consider the points $A(2,1),$ $B(10,6),$ $C(13,4),$ and $D(16,\mathrm{-2}).$ Determine the component form of vector $\overrightarrow{AD}.$

The speed of an object is the magnitude of its related velocity vector. A football thrown by a quarterback has an initial speed of $70$ mph and an angle of elevation of $30\text{°}.$ Determine the velocity vector in mph and express it in component form. (Round to two decimal places.)

A baseball player throws a baseball at an angle of $30\text{°}$ with the horizontal. If the initial speed of the ball is $100$ mph, find the horizontal and vertical components of the initial velocity vector of the baseball. (Round to two decimal places.)

A bullet is fired with an initial velocity of $1500$ ft/sec at an angle of $60\text{°}$ with the horizontal. Find the horizontal and vertical components of the velocity vector of the bullet. (Round to two decimal places.)

[T] A 65-kg sprinter exerts a force of $798$ N at a $19\text{°}$ angle with respect to the ground on the starting block at the instant a race begins. Find the horizontal component of the force. (Round to two decimal places.)

[T] Two forces, a horizontal force of $45$ lb and another of $52$ lb, act on the same object. The angle between these forces is $25\text{°}.$ Find the magnitude and direction angle from the positive x -axis of the resultant force that acts on the object. (Round to two decimal places.)

[T] Two forces, a vertical force of $26$ lb and another of $45$ lb, act on the same object. The angle between these forces is $55\text{°}.$ Find the magnitude and direction angle from the positive x -axis of the resultant force that acts on the object. (Round to two decimal places.)

[T] Three forces act on object. Two of the forces have the magnitudes $58$ N and $27$ N, and make angles $53\text{°}$ and $152\text{°},$ respectively, with the positive x -axis. Find the magnitude and the direction angle from the positive x -axis of the third force such that the resultant force acting on the object is zero. (Round to two decimal places.)

Three forces with magnitudes $80$ lb, $120$ lb, and $60$ lb act on an object at angles of $45\text{°},$ $60\text{°}$ and $30\text{°},$ respectively, with the positive x -axis. Find the magnitude and direction angle from the positive x -axis of the resultant force. (Round to two decimal places.)

[T] An airplane is flying in the direction of $43\text{°}$ east of north (also abbreviated as $\text{N}43\text{E})$ at a speed of $550$ mph. A wind with speed $25$ mph comes from the southwest at a bearing of $\text{N}15\text{E}.$ What are the ground speed and new direction of the airplane?

[T] A boat is traveling in the water at $30$ mph in a direction of $\text{N}20\text{E}$ (that is, $20\text{°}$ east of north). A strong current is moving at $15$ mph in a direction of $\text{N}45\text{E}.$ What are the new speed and direction of the boat?

[T] A 50-lb weight is hung by a cable so that the two portions of the cable make angles of $40\text{°}$ and $53\text{°},$ respectively, with the horizontal. Find the magnitudes of the forces of tension ${\text{T}}_1$ and ${\text{T}}_2$ in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

[T] A 62-lb weight hangs from a rope that makes the angles of $29\text{°}$ and $61\text{°},$ respectively, with the horizontal. Find the magnitudes of the forces of tension ${\text{T}}_1$ and ${\text{T}}_2$ in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

[T] A 1500-lb boat is parked on a ramp that makes an angle of $30\text{°}$ with the horizontal. The boat’s weight vector points downward and is a sum of two vectors: a horizontal vector ${\mathbf{\text{v}}}_1$ that is parallel to the ramp and a vertical vector ${\mathbf{\text{v}}}_2$ that is perpendicular to the inclined surface. The magnitudes of vectors ${\mathbf{\text{v}}}_1$ and ${\mathbf{\text{v}}}_2$ are the horizontal and vertical component, respectively, of the boat’s weight vector. Find the magnitudes of ${\mathbf{\text{v}}}_1$ and ${\mathbf{\text{v}}}_2.$ (Round to the nearest integer.)

[T] An 85-lb box is at rest on a $26\text{°}$ incline. Determine the magnitude of the force parallel to the incline necessary to keep the box from sliding. (Round to the nearest integer.)

A guy-wire supports a pole that is $75$ ft high. One end of the wire is attached to the top of the pole and the other end is anchored to the ground $50$ ft from the base of the pole. Determine the horizontal and vertical components of the force of tension in the wire if its magnitude is $50$ lb. (Round to the nearest integer.)

A telephone pole guy-wire has an angle of elevation of $35\text{°}$ with respect to the ground. The force of tension in the guy-wire is $120$ lb. Find the horizontal and vertical components of the force of tension. (Round to the nearest integer.)