a) Determine if the line [x,y,z] = [4,1,-2] +t[3,1,-5] has any x, y, or z-intercepts.
b) Under what conditions will a line parallel to [x,y,z] = [4,1,-2] +t[3,1,-5] have only
i. an x-intercept
ii. a y-intercept
iii. a z-intercept
c) Under what conditions (if any) will a line parallel to the one given have any x,y, or z - intercept?
a) It does have intercepts with x=0, y=0 and z=0 planes. In each case, a t-value exists.
b) A line parallel to
[x,y,z] = [4,1,-2] +t[3,1,-5]
would have to be of the form
[x,y,z] = [a,b,c] +t[3,1,-5]
It could not be parallel to x=0, y=0 or z=0 planes because, regardless of a,b or c, a t value could always be found to cross either of the three planes.
The answer is "none"
c) A line parallel to
[x,y,z] = [4,1,-2] +t[3,1,-5]
will always have x, y and z intercepts
a) To determine if the line [x,y,z] = [4,1,-2] + t[3,1,-5] has any x, y, or z-intercepts, we need to find the values of t that make each coordinate equal to zero.
Setting x = 0, we have:
0 = 4 + 3t
3t = -4
t = -4/3
Setting y = 0, we have:
0 = 1 + t
t = -1
Setting z = 0, we have:
0 = -2 - 5t
5t = -2
t = -2/5
Thus, the line has an x-intercept at t = -4/3, a y-intercept at t = -1, and a z-intercept at t = -2/5.
b) To determine the conditions for a line parallel to [x,y,z] = [4,1,-2] + t[3,1,-5] to have only an x-intercept, y-intercept, or z-intercept, we need to set one of the coefficients t in the direction vector [3,1,-5] to 0.
i. For the line to have only an x-intercept, we set the coefficient of t in the x-component to 0. This means the line is parallel to the yz-plane. So, the line is of the form [x,y,z] = [4,1,-2] + t[0,1,-5] or simply x = 4.
ii. For the line to have only a y-intercept, we set the coefficient of t in the y-component to 0. This means the line is parallel to the xz-plane. So, the line is of the form [x,y,z] = [4,1,-2] + t[3,0,-5] or simply y = 1.
iii. For the line to have only a z-intercept, we set the coefficient of t in the z-component to 0. This means the line is parallel to the xy-plane. So, the line is of the form [x,y,z] = [4,1,-2] + t[3,1,0] or simply z = -2.
c) If a line is parallel to the given line, it will have intercepts in any coordinate direction only if the corresponding coefficient of t in the direction vector is non-zero. This means a line parallel to the one given will have x-intercepts if the coefficient of t in the x-component is non-zero, y-intercepts if the coefficient of t in the y-component is non-zero, and z-intercepts if the coefficient of t in the z-component is non-zero.
To determine if the line [x, y, z] = [4, 1, -2] + t[3, 1, -5] has any x, y, or z-intercepts, we need to find when each component (x, y, and z) equals zero.
a) X-intercept: To find the x-intercept, set y and z to zero and solve for x.
Setting y and z to zero, we have:
x = 4 + 3t
0 = 4 + 3t
3t = -4
t = -4/3
So the x-coordinate will be zero when t = -4/3. Substituting this value of t back into the equation, we get:
x = 4 + 3(-4/3)
x = 4 - 4
x = 0
Therefore, the line intersects the x-axis at (0, 1, -2). It has an x-intercept.
b) To find the conditions for a line parallel to [x, y, z] = [4, 1, -2] + t[3, 1, -5] to have only an x-intercept, y-intercept, or z-intercept, we need to look at each component individually.
i. X-intercept: To have only an x-intercept, the y and z components of the direction vector ([3, 1, -5]) should be zero. So the condition is when the line is parallel to the x-axis.
ii. Y-intercept: To have only a y-intercept, the x and z components of the direction vector should be zero. So the condition is when the line is parallel to the y-axis.
iii. Z-intercept: To have only a z-intercept, the x and y components of the direction vector should be zero. So the condition is when the line is parallel to the z-axis.
c) To find the conditions for a line parallel to the given line to have any x, y, or z-intercept, any combination of the x, y, and z components of the direction vector can be nonzero. In other words, there are no specific conditions for this case. Any direction vector with nonzero x, y, or z components will result in a line with potential intercepts in all three dimensions.