A student solved the scalar equations for different planes as stated below. Explain why each equation is not the best representation for the equation for a plane. (4)
(a) 2x+4y−6z+24=0
(b) −2x+y−2z+5=0
for a is it because you can factor a 2 out of the equation?
and for b is it because of the negative infront of the equation? and can it be fixed by factoring a -1 out of the equation?
I agree, the first can be reduced to
x + 2y - 3z + 12 = 0
for the 2nd, it is normal practise to start the equation with a positive coefficient of x, so
2x - y + 2z - 5 = 0
To determine the best representation for the equation of a plane, we usually look for the simplified equation that follows certain conventions. Let's analyze each equation given and explain why they may not be the best representations.
(a) 2x + 4y − 6z + 24 = 0:
In this equation, you are correct that you can factor out a 2 from the equation, resulting in:
x + 2y - 3z + 12 = 0.
Reasons why it may not be the best representation:
1. Coefficients: The coefficients of the variables in the equation should preferably be integers or fractions. By factoring out a 2, we achieve this requirement, so this equation is an improvement in that aspect.
2. Normal Vector: The best representation of a plane equation has the coefficients of x, y, and z as the components of a vector that is perpendicular to the plane, known as the normal vector. In this equation, the normal vector would be (1, 2, -3).
3. Simplified Form: Ideally, the equation should be as simplified as possible. By simplifying the equation, we have x + 2y - 3z + 12 = 0, which is simpler than the original version and satisfies this requirement.
(b) −2x + y − 2z + 5 = 0:
For this equation, you are also correct that a negative sign is in front of the equation. However, simply factoring out a -1 would not be enough to fix it. Let's see why.
Reasons why it may not be the best representation:
1. Coefficients: As mentioned earlier, the coefficients of the variables should preferably be integers or fractions. In this case, there is a negative sign in front of the equation, which complicates the presentation.
2. Normal Vector: The coefficients of x, y, and z should form a vector perpendicular to the plane. In this case, the normal vector would be (-2, 1, -2).
3. Simplified Form: Ideally, the equation should be simplified. Factoring out a -1 would not be enough to reach the desired simplified form.
To improve equation (b), we could multiply the entire equation by -1 to eliminate the negative sign and simplify the coefficients, resulting in:
2x - y + 2z - 5 = 0
To recap, the best representation for a plane equation should have integer or fraction coefficients, a normal vector with coefficients matching those of x, y, and z, and be in the most simplified form.