Find a Vector Equation When Given Two Points - Pre-Calculus
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What is the vector that connects the point 
to 
?
What is the vector that connects the point
The first step when solving for a vetor is to find the length of the vector. It can be helpful to visualize the system as a right triangle as seen below:
At this point, we can essentially solve the problem as if we're finding the hypotenuse and angle of a right triangle.
Using the Pythagorean Theorem to find the length of the vector we get:
Now we need to find the angle of the vector. We have all three sides of the triangle, so use the trig function that you are most comfortable with. The tangent function is used below because it uses sides that were given in the problem statement.
The first step when solving for a vetor is to find the length of the vector. It can be helpful to visualize the system as a right triangle as seen below:
At this point, we can essentially solve the problem as if we're finding the hypotenuse and angle of a right triangle.
Using the Pythagorean Theorem to find the length of the vector we get:
Now we need to find the angle of the vector. We have all three sides of the triangle, so use the trig function that you are most comfortable with. The tangent function is used below because it uses sides that were given in the problem statement.
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Find the vector that starts at point 
and ends at 
and its magnitude.
Find the vector that starts at point
To find the vector between two points, find the change between the points in the 
and 
directions, or 
and 
. Then 
. If it helps, draw a line from the starting point to the end point on a graph and look at the changes in each direction.
We see that 
and 
, so our vector is
To find a vectors magnitude, we sum up the squares of each component and take the square root:
To find the vector between two points, find the change between the points in the
We see that
To find a vectors magnitude, we sum up the squares of each component and take the square root:
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Find the vector equation of the line through the points:
and 
.
Find the vector equation of the line through the points:
The vector equation of the line through two points is the sum of one of the points and the direction vector between the two points scaled by a variable.
First we find the the direction vector by subtracting the two points:
.
Note that a line is continuous and defined on the real line. Then, we must scale the direction vector by a variable constant so as to define the line at each point. We then add one of the given points, so as to define the line through the given points. Either point can be chosen, but the correct answer uses the first point given.
The vector equation of the line through two points is the sum of one of the points and the direction vector between the two points scaled by a variable.
First we find the the direction vector by subtracting the two points:
Note that a line is continuous and defined on the real line. Then, we must scale the direction vector by a variable constant so as to define the line at each point. We then add one of the given points, so as to define the line through the given points. Either point can be chosen, but the correct answer uses the first point given.
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Given points 
and 
, find a vector equation of the line passing these two points.
Given points
Write the formula to find the vector equation of the line.
Using 
and 
, find the directional vector 
by subtracting point A from B.
Substitute the directional vector 
and point 
into the formula.
A possible solution is: 
Write the formula to find the vector equation of the line.
Using
Substitute the directional vector
A possible solution is:
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