Geometric Vectors Practice Problems

Given the following vector, v, obtain the unit vector (u) that aligns with the direction of  v.

v = 13i

Given the following vector, v, obtain the unit vector (u) that aligns with the direction of  v.

v = 16 i - 12 j

Given the following vector, v, obtain the unit vector (u) that aligns with the direction of  v.

v = 11 i - 7 j

Given the following vector, v, obtain the unit vector (u) that aligns with the direction of  v.

v = 5 i + j

For the vector with the following magnitude || v|| and angle Q (in degrees), express the vector  v in terms of  i and j.

||v|| = 26, Q = 60°

For a vector v with the following magnitude || v|| and angle Q (in degrees), express it in terms of  i and j.

||v|| = 16, Q = 315°

For a vector v with the following magnitude ||v|| and angle Q, express it in terms of  i and j. Round values to four significant figures if necessary.

||v|| = 6/7, Q = 167°

Draw the following vector and consider the origin as its initial point. Then, calculate its magnitude.

v = 12i + 3j

Draw the following vector and consider the origin as its initial point. Then, calculate its magnitude.

v = 2i - 8j

Draw the following vector and consider the origin as its initial point. Then, calculate its magnitude.

v = -13i - 5j

Draw the following vector and consider the origin as its initial point. Then, calculate its magnitude.

v = -14i

A vector b starts at M and ends at N. Express the vector b in terms of i and j.

M = (-10, -3), N = (9, 4)

A vector b starts at M and ends at N. Express the vector b in terms of i and j.

M = (-12, 5), N = (-8, 1)

A vector b starts at M and ends at N. Express the vector b in terms of i and j.

M = (-3, 15), N = (-4, -4)

A vector b starts at M and ends at N. Express the vector b in terms of i and j.

M = (-10, 7), N = (14, 8)

For the given vectors a and b, find the magnitude of a and b. 

For the given vectors a and b, find out if a = b where a and b have the same direction.

For the given vector b, find the magnitude. 

For the given vectors a and b, find the magnitude of a and b. 

For the given vectors a and b, find out if a = b where a and b have the same direction.

For the given vector b, find the magnitude.

Considering the vectors a = 11i - 4j and b = -8i + 6j, determine the following vector:

a + b

Considering the vectors a = 11i - 4j and b = -8i + 6j, determine the following vector:

a - b

Considering the vectors a = 11i - 4j and b = -8i + 6j, determine the following vector:

b - a

Considering the vector a = 11i - 4j, determine the following vector:

30a

Considering the vector c = -9i + 6j, determine the following vector:

-21c

Considering the vectors b =  -12i - 8j and c = -9i + 6j, determine the following vector:

5c + 7b

Considering the vectors b =  -12i - 8j and c = -9i + 6j , determine the following vector:

9b - 15c

Considering the vector a = 11i - 4j, determine the following scalar:

||5a||

Considering the vectors a = 11i - 4j and b = -8i + 6j, determine the following scalar:

||b - a||

Find 7a - (3b - 2c) using the following vectors:

a = -5i + 6j, b = 2i + 7j, c = 10i + 9j

Find ||a + b||² - ||a - b||² using the following vectors:

a = -5i + 6j, b = 2i + 7j

Find the magnitude and direction angle of the vector "a" given below. Express your answer to one decimal place.

a = -14i + 21j

Find the magnitude and direction angle of the vector "a" given below. Express your answer to one decimal place.

a = (7i - 3j) - (7i - 13j)

Draw the following vector and consider the origin as its initial point. Then, calculate its magnitude.

v = -5i + 12j

Draw the following vector and consider the origin as its initial point. Then, calculate its magnitude.

v = -5i

The vector b starts from M and ends at N. Express the vector b in terms of i and j.

M = (-8, 0), N = (-7, -7)

Find 5a - 2b using the following vectors:

a = i - 11j and b = -3i + 9j

Find ||-7a|| using the following vector:

a = i - 11j

Given the following vector, a, obtain the unit vector that aligns with the direction of a.

a = -2i + 9j

The vector b starts from M and ends at N. Express the vector b in terms of i and j.

M = (-6, 7), N = (-3, 10)

Find 7a - 11b using the following vectors:

a = -3i + 8j and b = 8i - 13j

For the given vector, calculate its magnitude and direction angle. Express your answer to the nearest tenth, if required.

$〈11, 14〉$

For the given vector, calculate its magnitude and direction angle. Express your answer to the nearest tenth, if required.

$〈17, -13〉$

For the given vector, calculate its magnitude and direction angle. Express your answer to the nearest tenth, if required.

$〈-9,9\sqrt5〉$

The magnitude and direction angle of vector u are given below. Determine the horizontal and vertical components of the vector.

$\theta =74°,\mid \text{u}\mid =33$

The magnitude and direction angle of vector u are given below. Determine the horizontal and vertical components of the vector.

$\theta =42°,\mid \text{u}\mid =18.6$

Considering the vectors $\text{\textbf{a}}=〈-6,11〉$ and $\text{\textbf{b}}=〈13,7〉$ , determine the following vector:

$\text{\textbf{a}}-\text{\textbf{b}}$

Considering the vector $\text{\textbf{a}}=〈-6,11〉$, determine the following vector:

$-8\text{a}$

Considering the vectors $\text{\textbf{a}}=〈-6,11〉$ and $\text{\textbf{b}}=〈13,7〉$, determine the following vector:

$-8\textbf{a} + 11\textbf{b}$

Express the following vector into terms of $\textbf{i}$ and $\textbf{j}$ .

$〈9, -11〉$

Express the following vector into terms of $\textbf{i}$ and $\textbf{j}$ .

$〈13, 0〉$

For the following vectors, find the dot product.

$〈13, -4〉, 〈15, 8〉$

For the following vectors, find the dot product.

$〈9, 8〉, 〈-17, -16〉$

For the following vectors, find the dot product.

$11\text{i},17\text{i}-21\text{j}$

A vector is shown in the figure. Write it in the form 〈a, b〉. Express your answers as exact values or round them to four decimal places, if necessary.

A vector is shown in the figure. Write it in the form 〈a, b〉. Express your answers as exact values or round them to four decimal places, if necessary.

A vector is shown in the figure. Write it in the form 〈a, b〉. Express your answers as exact values or round them to four decimal places, if necessary.

A vector is shown in the figure. Write it in the form 〈a, b〉. Express your answers as exact values or round them to four decimal places, if necessary.

A vector is shown in the figure. Write it in the form 〈a, b〉. Express your answers as exact values or round them to four decimal places, if necessary.

Considering the vector $\textbf{a}=13\textbf{i}$ , determine the following vector:

$16\text{\textbf{a}}$

Considering the vectors $\textbf{a}=13\textbf{i}$ and $\textbf{b}=7\textbf{i}+17\textbf{j}$ , determine the following vector:

$7\textbf{a} + 12\textbf{b}$

Considering the vectors $\textbf{a}=13\textbf{i}$ and $\textbf{b}=7\textbf{i}+17\textbf{j}$ , determine the following vector:

$\textbf{a} - 10\textbf{b}$

Considering the vector $\text{\textbf{a}}=〈-11,17〉$ , determine the following vector:

$16\text{\textbf{a}}$

Considering the vectors $\text{\textbf{a}}=〈-11,17〉$ and $\text{\textbf{b}}=〈14,0〉$, determine the following vector:

$7\textbf{a} + 12\textbf{b}$

Considering the vectors $\text{\textbf{a}}=〈-11,17〉$ and $\text{\textbf{b}}=〈14,0〉$, determine the following vector:

$\textbf{a} - 10\textbf{b}$

Two vectors are shown in the figure. Find $\textbf{u}+\textbf{v}$. Express your answer in the form 〈a, b〉.

Two vectors are shown in the figure. Find $\textbf{u}-\textbf{v}$. Express your answer in the form 〈a, b〉.

For the following vectors, Find the value of the following expression $(8\text{\textbf{u}}) • \text{\textbf{v}}$ .

$\text{\textbf{u}}=〈-10,6〉$ and $\text{\textbf{v}}=〈12,19〉$

For the following vectors, Find the value of the following expression $\text{\textbf{u}} • \text{\textbf{v}} + \text{\textbf{u}} • \text{\textbf{w}}$ .

$\text{\textbf{u}}=〈-10,6〉$, $\text{\textbf{v}}=〈12,19〉$ and $\text{\textbf{w}}=〈15,4〉$

Refer to the vector shown below. Sketch the vector as indicated.

$-l$

Refer to the vector shown below. Sketch the vector as indicated.

5m

A vector is shown in the figure. Find $-\textbf{u}$. Express your answer in the form 〈a, b〉.

Two vectors are shown in the figure. Find $\textbf{u}+\textbf{v}$. Express your answer in the form 〈a, b〉.

Two vectors are shown in the figure. Find $\textbf{u}-\textbf{v}$. Express your answer in the form 〈a, b〉.

A vector is shown in the figure. Find $-\textbf{v}$. Express your answer in the form 〈a, b〉.

Two vectors are shown in the figure. Find $\textbf{u}+\textbf{v}$. Express your answer in the form 〈a, b〉.

Two vectors are shown in the figure. Find $\textbf{u}-\textbf{v}$. Express your answer in the form 〈a, b〉.

A vector is shown in the figure. Find $-\textbf{u}$. Express your answer in the form 〈a, b〉.

Two vectors are shown in the figure. Find $\textbf{u}+\textbf{v}$. Express your answer in the form 〈a, b〉.

Two vectors are shown in the figure. Find $\textbf{u}-\textbf{v}$. Express your answer in the form 〈a, b〉.

A vector is shown in the figure. Find $-\textbf{v}$. Express your answer in the form 〈a, b〉.

Two bicycles are pulling a bicycle that has a broken chain. The forces exerted by the two bicycles are 12 lb, and 8 lb, and the angle between these forces is 35.6°. Determine the magnitude of the equilibrant and the angle it makes with the 12 lb force.

Refer to the vectors shown below. Sketch the vector as indicated. Use the parallelogram rule to get the resultant.

p + q

Two Siberian Huskies are pulling a sled on a snowy terrain. The forces exerted by these dogs on the sled are 16 lb, and 5 lb, and the angle between these forces is 18.3°. Determine the magnitude of the equilibrant and the angle it makes with the 16 lb force.

Two forces, 898 newtons, and 356 newtons, are concurrent (their lines of action pass through a common point). If the resultant force is 1044 newtons, what is the angle between the two forces?

Two forces, 428 lb, and 729 lb, are concurrent (their lines of action pass through a common point). If the resultant force is 986 lb, what is the angle between the two forces?

Calculate the magnitude of the resultant force. Write your answer in one decimal place.

Calculate the magnitude of the resultant force. Write your answer in one decimal place.

Two forces of 180 and 350 newtons act on a point in the passenger train. Calculate the magnitude of the resultant force if the angle between them is $78°$​​.

Two forces of 108 and 128 lb act on a point in the passenger train. Calculate the magnitude of the resultant force if the angle between them is 155° 40'​​.

A person is holding an 85.0 lb trolly bag on the ramp outside an airport. Find the angle of the ramp with the ground if the bag experiences a force of 29.0 lb.

Robert takes out his baby for an evening walk. The baby is sitting on a tricycle. Robert is holding a 100.0 lb baby's cycle with a force of 50.0 lb outside the ramp of his house. Find the angle of inclination of the ramp with the ground.

Refer to the vectors shown below. Sketch the vector as indicated. Use the parallelogram rule to get the resultant.

p - r

Two forces A and B are shown and are acting on a certain point. The angle between forces A and B is 63° 40'. Determine the magnitude of force B and the resultant if the resultant makes an angle of 27° 20' with force A and the magnitude of force A is 812 lb.

Two forces A and B are shown and are acting on a certain point. The angle between forces A and B is 72° 30'. Determine the magnitude of force B and the resultant if the resultant makes an angle of 65° 50' with force A and the magnitude of force A is 81.6 lb.

A 14-lb force is needed to keep the box filled with Trigonometry books stable on an inclined ramp. If the weight of the box is 46 lb, what is the angle between the ramp and the horizontal?

A garbage truck is parked on a hilly road. If the weight of the truck is 7200 lb and the angle between the road and the horizontal is 21°, what is the force required to keep the truck on its position?

A cylindrical container filled with chemicals is being transported on a conveyor in a factory. A segment of the conveyor has a length of 30 ft and is inclined at an angle of 9.2° with the horizontal. If the container weighs 173 lb, what force is required to hold it as it passes through the inclined segment?

Refer to the vectors shown below. Sketch the vector as indicated. Use the parallelogram rule to get the resultant.

p + (q + r)

Refer to the vectors shown below. Sketch the vector as indicated. Use the parallelogram rule to get the resultant.

r + s

Draw the resultant vector for the pair of vectors with angle α between them using the given information.

|a| = 7, |b| = 15, α = 54°

Draw the resultant vector for the pair of vectors with angle α between them using the given information.

|a| = 17, |b| = 25, α = 34°

Two porters are carrying a luggage box of a passenger. The first porter exerts a force of 180 lb at an angle of $53.2°$ while the second porter exerts a force of 111 lb at an angle of $72.8°$. Calculate the weight of the box.

A trunk full of fruits and vegetables is tied by two strings in the opposite direction. The first string makes an angle of $38\degree30^{\prime}$ with the ground and has a tension of 76.5 lb. The other string is parallel to the ground. Calculate the weight of the trunk and tension in the other string.

A ship sails 19.6 miles out of port with a bearing of 321°. After a few hours, the ship makes a turn and starts moving to the west for 8.3 miles. Find the bearing of the ship and its distance from the port.

A motorboat departs 10.4 miles out of port with a bearing of 313°. After a few hours, the motor makes a turn and starts moving to the west for 6.9 mi. Find the bearing of the ship and its distance from the port.

A luxury ship starts from point P and covers a distance of 21.7 km with a bearing of $199°$. After reaching point Q, the ship turns back and covers a distance of 65.2 km with a bearing of $331°$. Calculate the distance of the ship from the starting point.

A luxury ship starts from point P and covers a distance of 12.9 km with a bearing of $209°$. After reaching point Q, the ship turns back and covers a distance of 5.8 km with a bearing of $347°$ . Calculate the distance of the ship from the starting point.

A yacht is sailing in the direction S 59° 00′ E. If the speed of the current is zero, the yacht sails at a speed of 37 km/h. The resultant direction of the yacht is in the east because the current is flowing directly in the north. Determine the actual speed of the yacht and the speed of the current.

Determine the ground speed of a charter plane heading from NY to LA on a bearing of 257.0°. The air speed of the charter plane is 550 mph. It encounters wind currents in the direction of 125° at 50 mph. Also, determine the resultant bearing of the plane.

A charter departs from London, heading at a speed of 550 mph from the direction of 169.2°. The wind is blowing against the charter at a speed of 30 mph, in the direction of 263.9°. What is the resulting bearing of the charter?

Sam is flying an aerobatic plane on a windy day. The wind's speed is 30 mph and blowing from the south. The plane is flying at a speed of 212 mph and Sam wants its flying path to be on a bearing of 61° 20'. Determine the ground speed of the plane and the bearing he should fly to stay on that path.

Determine the ground speed of a fighter jet that is headed north at an airspeed of 525 km/h if the wind is blowing from a direction of 69° at 70 km/h. Also, find the resulting bearing of the jet.

Determine the angle between the given vectors and state if the vectors are perpendicular or not. Write the answer rounded to the nearest tenth of a degree. <-2, 5>, <5, 2>

A rope is strangled on a pole. One end of the rope is pulled by Kid A with a force of 16 lb and another end of the rope is pulled by Kid B with a force of 22 lb at an angle of 60° from the first force. Determine the resultant force applied on the pole. Also, find out the angle between the resultant and the force applied by the kid A.