Any ideas on how to solve this: Use a vector solution to show that a scalar equation for the line through the points P1(x1, y1) and P2(x2, y2) is y-y1/x-x1 = y2-y1/x2-x1? Very confused as to what is meant by this question. Thanks.
This just says, correctly, thst the change in y from point (x1,y1) is equal to the slope times the change in x from that point.
The slope of the line is (y2-y1)/(x2-x1)
To solve this problem, you need to use vectors to show that the given scalar equation represents the line passing through points P1(x1, y1) and P2(x2, y2). Here's a step-by-step solution:
Step 1: Define the position vectors of P1 and P2
Let vector A represent the position vector of point P1 and vector B represent the position vector of point P2. We can express these vectors as follows:
A = (x1, y1)
B = (x2, y2)
Step 2: Find the displacement vector between P1 and P2
The displacement vector between two points is calculated by subtracting the initial position vector from the final position vector. Therefore, the displacement vector (AB) between P1 and P2 can be obtained as follows:
AB = B - A = (x2 - x1, y2 - y1)
Step 3: Express the given scalar equation using vector notation
The given scalar equation can be written as:
(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)
To express this equation using vector notation, make the following substitutions:
(y - y1) → v
(x - x1) → u
(y2 - y1) → b
(x2 - x1) → a
The equation now becomes:
v / u = b / a
Step 4: Show the vector equation of the line
To show that the equation represents the line passing through P1 and P2, express the vectors in terms of the displacement vector AB:
v = (y - y1) = (AB.y / AB.x) * u
b = (y2 - y1) = (AB.y / AB.x) * a
Step 5: Compare the vector equation with the given scalar equation
If you compare the vector equation with the given scalar equation, you can see that they both have the same form:
v / u = b / a
This confirms that the scalar equation represents the line passing through P1 and P2.
By following these steps, you have shown that the given scalar equation is equivalent to the vector equation representing the line passing through P1(x1, y1) and P2(x2, y2).
To solve this problem, we will use the concept of vectors. Here's a step-by-step explanation on how to approach this:
Step 1: Define the position vectors
Let P1 denote the position vector of point P1(x1, y1), and P2 denote the position vector of point P2(x2, y2). Each position vector is determined by subtracting the origin (0, 0) from the respective point coordinates.
P1 = (x1, y1) - (0, 0) = (x1, y1)
P2 = (x2, y2) - (0, 0) = (x2, y2)
Step 2: Find the direction vector
The direction vector of the line passing through P1 and P2 is obtained by subtracting the position vector of P1 from that of P2.
D = P2 - P1
= (x2, y2) - (x1, y1)
= (x2 - x1, y2 - y1)
Step 3: Express the scalar equation using the direction vector
Now, let's express the equation of the line as given: y - y1 / x - x1 = y2 - y1 / x2 - x1.
Since we have the direction vector D, we can rewrite this equation as:
(y - y1) / (x - x1) = (D2) / (D1)
Note that D1 represents the x-component of the direction vector, and D2 represents the y-component.
Step 4: Simplify the equation using the direction vector
Now, we can substitute the values of the direction vector into the equation:
(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)
This equation shows that the line passing through the points P1 and P2 can be represented by a scalar equation.
In summary, the solution involves determining the position vectors P1 and P2, finding the direction vector D by subtracting P1 from P2, and then substituting the values from the direction vector into the given scalar equation for the line.