Given the vector p=i+2j-2k and q=2i-j+2k. Find two vectors m and n satisfyin all the followin conditions:m is pendicular to p , n is pendicular to p,m+n=q.
M= 1/9(22î-j+10k)
To find two vectors, m and n, that satisfy the given conditions, we proceed as follows:
1. Determine the vector m that is perpendicular to p:
To find m, we need to take the cross product of p and another vector that is perpendicular to it. Let's choose the vector u = j+k as a candidate for a vector perpendicular to p.
The cross product of p and u is given by:
m = p x u = (i + 2j - 2k) x (0i + 1j + 1k)
= 2i + 2k
Therefore, the vector m that is perpendicular to p is m = 2i + 2k.
2. Determine the vector n that is perpendicular to p:
Since n is also perpendicular to p, we can find it similarly to how we found m.
Let's choose the vector v = i+k as another candidate for a vector perpendicular to p.
The cross product of p and v is given by:
n = p x v = (i + 2j - 2k) x (1i + 0j + 1k)
= -4j + 3k
Therefore, the vector n that is perpendicular to p is n = -4j + 3k.
3. Verify that the sum of m and n is equal to vector q:
Now, let's check if the sum of m and n equals vector q.
m + n = (2i + 2k) + (-4j + 3k)
= 2i - 4j + 2k + 3k
= 2i - 4j + 5k
Comparing this result with q = 2i - j + 2k, we can conclude that m + n is indeed equal to q.
Therefore, the vectors m and n that satisfy all the given conditions are:
m = 2i + 2k
n = -4j + 3k
To find two vectors, m and n, that satisfy the given conditions, we can use the properties of dot products and vector addition.
First, let's find a vector m that is perpendicular to p.
To determine the direction of m, we can take the cross product of p and another vector, say v. The cross product of two vectors is perpendicular to each of the vectors involved.
Let's assume v=a*i+b*j+c*k, where a, b, and c are constants.
To find m, we need the cross product of p and v to be zero (since m needs to be perpendicular to p).
The cross product of two vectors can be calculated using the determinant formula:
m = (p x v) = |i j k|
|1 2 -2|
|a b c|
Expanding this determinant, we get:
m = (2b + 2c)i - (a - 2c)j - (2a + b)k
Since m needs to be perpendicular to p, the dot product of m and p must be zero:
m · p = (2b + 2c) * 1 + (a - 2c) * 2 + (2a + b) * (-2) = 0
Simplifying this equation, we get:
2a + 2b + 6c = 0 (Equation 1)
Now, let's find a vector n that is perpendicular to p and satisfying the condition m + n = q.
Since n is perpendicular to p, we again assume n = x*i + y*j + z*k, where x, y, and z are constants.
n + m = q
(x*i + y*j + z*k) + ((2b + 2c)i - (a - 2c)j - (2a + b)k) = (2i - j + 2k)
This gives us the following equations:
x + 2b + 2c = 2 (Equation 2)
y + (a - 2c) = -1 (Equation 3)
z - (2a + b) = 2 (Equation 4)
We now have three equations (Equations 1, 2, and 3) and three unknowns (a, b, and c). By solving this system of equations, we can find the values of a, b, and c, which will then lead to the values of x, y, and z.
After obtaining the values of x, y, and z, we have our vectors m and n satisfying all the given conditions.