AP Calculus BC › Vectors and Vector Operations
Compute the following: .
The formula for subtracting vectors is . Plugging in the values we were given, we get
Find the dot product of the vectors and
To find the dot product of two vectors and
, we use the formula
Using the vectors from the problem statement, we get
Find the dot product between the vectors and
To find the dot product between two vectors and
, we apply the following formula:
Using the vectors from the problem statement, we get
Determine whether the two vectors, and
, are orthogonal or not.
The two vectors are not orthogonal.
The two vectors are orthogonal.
Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:
To find the dot product of two vectors given the notation
Simply multiply terms across rows:
For our vectors, and
The two vectors are not orthogonal.
Find the dot product between the vectors and
To find the dot product between two vectors and
, we apply the following formula:
Using the vectors from the problem statement, we get
Find the dot product of the vectors and
To find the dot product of two vectors and
, we apply the formula
Using the vectors from the problem statement, we get
Given the following two vectors, and
, calculate the dot product between them,
.
The dot product of a paired set of vectors can be found by summing up the individual products of the multiplications between matched directional vectors.
Note that the dot product is a scalar value rather than a vector; there's no directional term.
Now considering our problem, we're given the vectors and
The dot product can be found following the example above:
Find the dot product between the vectors and
To find the dot product between the vectors and
, we use the formula
Using the vectors from the problem statement, we then get
Find the dot product between the vectors and
To find the dot product between the vectors and
, we use the formula
. Using the vectors from the problem statement, we get
Find the angle between the following vectors (to two decimal places):
1.46
1.18
0.98
2.89
1.62
The dot product is defined as:
Where theta is the angle between the two vectors. Solving for theta:
To solve each component:
Putting it all together, we can solve for theta: