All Calculus 3 Resources
Example Questions
Example Question #1 :Angle Between Vectors
Find the angle between these two vectors,, and.
Lets remember the formula for finding the angle between two vectors.
Example Question #2 :Angle Between Vectors
Calculate the angle between,.
Lets recall the equation for finding the angle between vectors.
Example Question #3 :Angle Between Vectors
What is the angle between the vectorsand?
To find the angle between vectors, we must use the dot product formula
whereis the dot product of the vectorsand, respectively.
andare the magnitudes of vectorsand, respectively.
is the angle between the two vectors.
Let vectorbe represented asand vectorbe represented as.
The dot product of the vectorsandis.
The magnitude of vectorisand vectoris.
Rearranging the dot product formula to solve forgives us
For this problem,
Example Question #4 :Angle Between Vectors
What is the angle between the vectorsand?
To find the angle between vectors, we must use the dot product formula
whereis the dot product of the vectorsand, respectively.
andare the magnitudes of vectorsand, respectively.
is the angle between the two vectors.
Let vectorbe represented asand vectorbe represented as.
The dot product of the vectorsandis.
The magnitude of vectorisand vectoris.
Rearranging the dot product formula to solve forgives us
For this problem,
The vectors are perpendicular
Example Question #5 :Angle Between Vectors
What is the angle between the vectorsand?
To find the angle between vectors, we must use the dot product formula
whereis the dot product of the vectorsand, respectively.
andare the magnitudes of vectorsand, respectively.
is the angle between the two vectors.
Let vectorbe represented asand vectorbe represented as.
The dot product of the vectorsandis.
The magnitude of vectorisand vectoris.
Rearranging the dot product formula to solve forgives us
For this problem,
Example Question #6 :Angle Between Vectors
What is the angle between the vectorsand?
To find the angle between vectors, we must use the dot product formula
whereis the dot product of the vectorsand, respectively.
andare the magnitudes of vectorsand, respectively.
is the angle between the two vectors.
Let vectorbe represented asand vectorbe represented as.
The dot product of the vectorsandis.
The magnitude of vectorisand vectoris.
Rearranging the dot product formula to solve forgives us
For this problem,
The two vectors are parallel.
Example Question #7 :Angle Between Vectors
Find the approximate acute angle in degrees between the vectors.
None of the other answers
To find the angle between two vectors, use the formula
.
Example Question #8 :Angle Between Vectors
Find the angle between the following two vectors.
In order to find the angle between two vectors, we need to take the quotient of their dot product and their magnitudes:
Therefore, we find that
.
Example Question #9 :Angle Between Vectors
Find the (acute) angle between the vectorsin degrees.
To find the angle between vectors, we use the formula
.
Substituting in our values, we get
Example Question #10 :Angle Between Vectors
Find the angle between the two vectors.
No angle exists
To find the angle between two vector we use the following formula
and solve for.
鉴于
we find
Plugging these values in we get
To findwe calculate theof both sides
and find that
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