Question

Denote by 𝑄𝑅 and 𝑆 the projections of the point P=(−2,3,−4) onto the 𝑥𝑦 plane, the 𝑦𝑧 plane, and the 𝑥𝑧 plane, respectively. Which of the following line segments has the greatest length?

Answer 1

Solution:

We know that $xy$ -plane is described by equation $z=0$ .

So, projection of $P(-2,3,-4)$ on $xy$ -plane is $Q(-2,3,0)$ .

We know that $yz$ -plane is described by equation $x=0$ .

So, projection of $P(-2,3,-4)$ on $yz$ -plane is $R(0,3,-4)$ .

We know that $xz$ -plane is described by equation $y=0$ .

So, projection of $P(-2,3,-4)$ on $xz$ -plane is $S(-2,0,-4)$ .

Now, we find their distances or length with $O(0,0,0)$ using distance formula.

$QO=\sqrt{(-2-0)^2+(3-0)^2+(0-0)^2}=\sqrt{4+9}=\sqrt{13}$

$RO=\sqrt{(0-0)^2+(3-0)^2+(-4-0)^2}=\sqrt{9+16}=\sqrt{25}$

$SO=\sqrt{(-2-0)^2+(0-0)^2+(-4-0)^2}=\sqrt{4+16}=\sqrt{20}$

Clearly, $\sqrt{25}$ is greater, thus RO is the longest length.