Geometric Vectors - Pre-Calculus
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Vector 
has a magnitude of 2.24 and is at an angle of 63.4° CCW from the x-axis. Vector 
has a magnitude of 3.16 at an anlge of 342° CCW from the x-axis.
Find 
by using the parallelogram graphical method.
Vector
Find
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First, construct the two vectors using ruler and protractor:
Place the tails of both vectors at the same point:
Construct a parallelogram:
Construct and measure the resultant using ruler and protractor:
First, construct the two vectors using ruler and protractor:
Place the tails of both vectors at the same point:
Construct a parallelogram:
Construct and measure the resultant using ruler and protractor:
Find the norm of the vector: 
Find the norm of the vector:
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The norm of a vector is also known as the length of the vector. The norm is given by the formula:
.
Here, we have
,
the correct answer.
The norm of a vector is also known as the length of the vector. The norm is given by the formula:
Here, we have
the correct answer.
Express a vector with magnitude 2.24 directed 63.4° CCW from the x-axis in unit vector form.
Express a vector with magnitude 2.24 directed 63.4° CCW from the x-axis in unit vector form.
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The x-coordinate is the magnitude times the cosine of the angle, while the y-coordinate is the magnitude times the sine of the angle.
The resultant vector is: 
.
The x-coordinate is the magnitude times the cosine of the angle, while the y-coordinate is the magnitude times the sine of the angle.
The resultant vector is:
What is the magnitude and angle for the following vector, measured CCW from the x-axis?
What is the magnitude and angle for the following vector, measured CCW from the x-axis?
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The magnitude of the vector is found using the distance formula:
To calculate the angle we must first find the inverse tangent of 
:
This angle value is the principal arctan, but it is in the fourth quadrant while our vector is in the second. We must add the angle 180° to this value to arrive at our final answer.
The magnitude of the vector is found using the distance formula:
To calculate the angle we must first find the inverse tangent of
This angle value is the principal arctan, but it is in the fourth quadrant while our vector is in the second. We must add the angle 180° to this value to arrive at our final answer.
Vector 
has a magnitude of 3.61 and a direction 124° CCW from the x-axis. Express 
in unit vector form.
Vector
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For vector 
, the magnitude is doubled, but the direction remains the same.
For our calculation, we use a magnitude of:
The x-coordinate is the magnitude times the cosine of the angle, while the y-coordinate is the magnitude times the sine of the angle.
The resultant vector is: 
.
For vector
For our calculation, we use a magnitude of:
The x-coordinate is the magnitude times the cosine of the angle, while the y-coordinate is the magnitude times the sine of the angle.
The resultant vector is:
What are the magnitude and angle, CCW from the x-axis, of 
?
What are the magnitude and angle, CCW from the x-axis, of
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When multiplying a vector by a constant (called scalar multiplication), we multiply each component by the constant.
The magnitude of this new vector is found with these new components:
To calculate the angle we must first find the inverse tangent of 
:
This is the principal arctan, but it is in the first quadrant while our vector is in the third. We to add the angle 180° to this value to arrive at our final answer.
When multiplying a vector by a constant (called scalar multiplication), we multiply each component by the constant.
The magnitude of this new vector is found with these new components:
To calculate the angle we must first find the inverse tangent of
This is the principal arctan, but it is in the first quadrant while our vector is in the third. We to add the angle 180° to this value to arrive at our final answer.
Vector 
has a magnitude of 2.24 and is at an angle of 63.4° CCW from the x-axis. Vector 
has a magnitude of 3.16 at an angle of 342° CCW from the x-axis.
Find 
by using the nose-to-tail graphical method.
Vector
Find
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First, construct the two vectors using ruler and protractor:
Place the tail of 
at the nose of 
:
Construct the resultant 
from the tail of 
to the nose of 
:
With our ruler and protractor, we find that 
is 4.12 at an angle of 14.0° CCW from the x-axis.
First, construct the two vectors using ruler and protractor:
Place the tail of
Construct the resultant
With our ruler and protractor, we find that
Find the magnitude and angle CCW from the x-axis of 
using the nose-to-tail graphical method.
Find the magnitude and angle CCW from the x-axis of
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Construct 
and 
from their x- and y-components:
Since we are subtracting, reverse the direction of 
:
Form 
by placing the tail of 
at the nose of 
:
Construct and measure the resultant, 
, from the tail of 
to the nose of 
using a ruler and protractor:
Construct
Since we are subtracting, reverse the direction of
Form
Construct and measure the resultant,
Vector 
has a magnitude of 2.24 and is at an angle of 63.4° CCW from the x-axis. Vector 
has a magnitude of 3.61 and is at an angle of 124° CCW from the x-axis.
Find 
by using the nose-to-tail graphical method.
Vector
Find
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First, construct the two vectors using ruler and protractor:
is twice the length of 
, but in the same direction:
Since we are subtracting, reverse the direction of 
:
Form 
by placing the tail of 
at the nose of 
:
Construct and measure the resultant 
from the tail of 
to the nose of 
with a ruler and protractor.
First, construct the two vectors using ruler and protractor:
Since we are subtracting, reverse the direction of
Form
Construct and measure the resultant
Find 
using the parallelogram graphical method.
Find
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Construct 
and 
from their x- and y-components:
Since we are subtracting, reverse the direction of 
:
Place the tails of 
and 
at the same point:
Construct a parallelogram:
Construct and measure the resultant using a ruler and protractor.
Construct
Since we are subtracting, reverse the direction of
Place the tails of
Construct a parallelogram:
Construct and measure the resultant using a ruler and protractor.
Vector 
has a magnitude of 3.61 and is at an angle of 124° CCW from the x-axis. Vector 
has a magnitude of 2.24 at an anlge of 63.4° CCW from the x-axis.
Find 
using the parallelogram graphical method.
Vector
Find
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First, construct the two vectors using ruler and protractor:
is twice the length of 
, but in the same direction:
Since we are subtracting, reverse the direction of 
:
Place the tails of 
and 
at the same point:
Construct a parallelogram:
Construct and measure the resultant using ruler and protractor:
First, construct the two vectors using ruler and protractor:
Since we are subtracting, reverse the direction of
Place the tails of
Construct a parallelogram:
Construct and measure the resultant using ruler and protractor:
Find 
.
Find
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Finding the resultant requires us to add like components:
Finding the resultant requires us to add like components:
Find 
.
Find
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Finding the resultant requires us to add like components:
Finding the resultant requires us to add like components:
Evaluate 
Evaluate
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When adding two vectors, they need to be expanded into their components. Luckily, the problem statement gives us the vectors already in their component form. From here, we just need to remember that we can only add like components. So for this problem we get:
Now we can combine those values to write out the complete vector:
When adding two vectors, they need to be expanded into their components. Luckily, the problem statement gives us the vectors already in their component form. From here, we just need to remember that we can only add like components. So for this problem we get:
Now we can combine those values to write out the complete vector:
Find the norm of the vector 
.
Find the norm of the vector
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We find the norm of a vector by finding the sum of each element squared and then taking the square root.
.
We find the norm of a vector by finding the sum of each element squared and then taking the square root.
Find the product of the vector 
and the scalar 
.
Find the product of the vector
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When multiplying a vector by a scalar we multiply each component of the vector by the scalar and the result is a vector:
When multiplying a vector by a scalar we multiply each component of the vector by the scalar and the result is a vector:
Find the norm of the vector 
.
Find the norm of the vector
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We find the norm of a vector by finding the sum of each component squared and then taking the square root of that sum.
We find the norm of a vector by finding the sum of each component squared and then taking the square root of that sum.
This question refers to the previous question.
Simplify.
This question refers to the previous question.
Simplify.
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In order to simplify this problem we need to multiply the scalar factor to each component of the vector.
In our case the scalar factor is 
Thus,
In order to simplify this problem we need to multiply the scalar factor to each component of the vector.
In our case the scalar factor is
Thus,
Find the vector given by the product: 
Find the vector given by the product:
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Given a scalar k and a vector v, the vector given by their products is defined component-wise:
.
Here, our product is:
Given a scalar k and a vector v, the vector given by their products is defined component-wise:
Here, our product is:
Find the norm of vector 
.
Find the norm of vector
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Write the formula to find the norm, or the length the vector.
Substitute the known values of the vector and solve.
Write the formula to find the norm, or the length the vector.
Substitute the known values of the vector and solve.