A vector equation for a given straight line is r = (i + 3j) + lambda (-i - j)
Construct a vector equation for the line that does go through the point (1,2), and is perpendicular to r.
and
Determine the point of intersection of the two lines
you know that i-j is perpendicular to -i-j since (-1)(1) + (-1)(-1) = -1+1 = 0
so, i+2j + alpha(i - j) will do the trick
1-lambda = 1 + alpha
3-lambda = 2 - alpha
intersect where lambda = 1/2 alpha = -1/2, at the point (1/2,5/2)
To find the vector equation for a line that passes through the point (1,2) and is perpendicular to the given line, we need to find the direction vector of the perpendicular line.
1. Direction vector of the given line:
The direction vector of the given line is (-i - j).
2. Perpendicular direction vector:
To find the perpendicular direction vector, we need to take the dot product of the direction vector of the given line with the perpendicular direction vector, which should be zero.
(-i - j) ⋅ (a i + b j) = 0
Simplifying the dot product: -a - b = 0
We can choose a = 1 and b = -1 to satisfy this equation.
3. Perpendicular line equation:
Using the point-slope form of a line (r = r0 + λv), where r0 is the position vector of the point (1,2) and v is the perpendicular direction vector, the equation becomes:
r = (1i + 2j) + λ(1i - 1j)
Simplifying the equation: r = (1 + λ)i + (2 - λ)j
Now, to determine the point of intersection of the two lines, we need to solve their equations simultaneously.
Given line equation: r = (i + 3j) + λ(-i - j)
Perpendicular line equation: r = (1 + λ)i + (2 - λ)j
Equating the two equations:
(i + 3j) + λ(-i - j) = (1 + λ)i + (2 - λ)j
Separating the coefficients of i and j:
i - λi = i + λ
3j - λj = 2j - λj
Simplifying the equations:
1 - λ = 1 + λ
3 - λ = 2 - λ
Solving for λ:
1 - λ = 1 + λ
2λ = 0
λ = 0
Substituting λ = 0 into either equation, we get:
1 - λ = 1 + λ
1 - 0 = 1 + 0
1 = 1
Therefore, the point of intersection of the two lines is (1,1).
To construct a vector equation for a line that goes through a given point and is perpendicular to a given line, follow these steps:
Step 1: Find the direction vector of the given line.
In this case, the direction vector of the given line is the coefficient of lambda in the given vector equation. It is -i - j.
Step 2: Find a vector perpendicular to the direction vector.
To find a vector perpendicular to -i - j, we can take the cross product of -i - j with the unit vector in the z-direction, which is k. So we have:
n = (-i - j) x k
= (-i - j) x (0i + 0j + 1k)
= -j i - (-i j)
= -i + j
Now, we have a vector n = -i + j perpendicular to the given line.
Step 3: Construct the vector equation using the point and the perpendicular vector.
Let (a, b) be the point that the line goes through. In this case, (a, b) = (1, 2).
The vector equation for the line is then:
R = (a + λn)i + (b + λn)j
= (1 + λ(-i + j))i + (2 + λ(-i + j))j
So, the vector equation for the line that goes through the point (1, 2) and is perpendicular to the given line is:
R = (1 - λ)i + (2 + λ)j
To determine the point of intersection of two lines, you need to equate their position vectors and solve the resulting system of equations. However, in this case, you have only provided one equation. So, we cannot determine the point of intersection with the information given.