Final answer:
The vector function r(t) = <3 cos t , 0, sin t> graphed in the xz-plane is not an ellipse, but a circle with a radius of 3 units on the x-axis and 1 unit on the z-axis, hence the statement is False.
Step-by-step explanation:
The question is asking whether the graph of the vector function r(t) = <3 cos t , 0, sin t> is an ellipse in the xz-plane. To analyze this, we can look at the individual components of the vector function, which represent the x and z positions of the graph over time, as there is no y-component since it is always zero.
The x-component is given by 3 cos t, and the z-component is given by sin t. To check if this forms an ellipse in the xz-plane, we need to see if it satisfies the standard form of an ellipse equation, which is (x/a)^2 + (y/b)^2 = 1 for some constants a and b, where y is always zero in this particular case.
If we square the x and z components, and divide them by their coefficients squared respectively, we get (cos t)^2 + (sin t)^2 = 1, which is the unit circle equation, not an ellipse (since an ellipse would require the coefficients of cos t and sin t to be different). Therefore, the statement is False; the graph of r(t) in the xz-plane is not an ellipse, but a circle with a radius of 3 units on the x-axis and 1 unit on the z-axis.