Math Problem Statement

Consider the line ℓ:r(t)=(8,7,3)+t(2,9,3),t∈Rℓ:r(t)=(8,7,3)+t(2,9,3),t∈R. Let P(x,y,z)P(x,y,z) be the point on the line ℓℓ that is nearest to the origin. Find x,yx,y and zz. Express your solution in exact form.

Solution

To find the point P(x,y,z)P(x, y, z) on the line \ell that is nearest to the origin, we need to minimize the distance from the origin to any point on the line.

The line \ell is given by the parametric equation:

r(t)=(8,7,3)+t(2,9,3),tRr(t) = (8, 7, 3) + t(2, 9, 3), \quad t \in \mathbb{R}

A point on the line can be represented as:

P(t)=(x(t),y(t),z(t))=(8+2t,7+9t,3+3t)P(t) = (x(t), y(t), z(t)) = (8 + 2t, 7 + 9t, 3 + 3t)

The distance DD from the origin to the point P(t)P(t) is given by:

D=x(t)2+y(t)2+z(t)2D = \sqrt{x(t)^2 + y(t)^2 + z(t)^2}

To minimize this distance, we minimize the squared distance D2D^2 (since the square root is a monotonic function, the minimum of D2D^2 will also minimize DD):

D2=(x(t))2+(y(t))2+(z(t))2D^2 = (x(t))^2 + (y(t))^2 + (z(t))^2

Substitute the expressions for x(t),y(t),z(t)x(t), y(t), z(t):

D2=(8+2t)2+(7+9t)2+(3+3t)2D^2 = (8 + 2t)^2 + (7 + 9t)^2 + (3 + 3t)^2

Now, expand and simplify the expression:

D2=(64+32t+4t2)+(49+126t+81t2)+(9+18t+9t2)D^2 = (64 + 32t + 4t^2) + (49 + 126t + 81t^2) + (9 + 18t + 9t^2)

D2=(64+49+9)+(32t+126t+18t)+(4t2+81t2+9t2)D^2 = (64 + 49 + 9) + (32t + 126t + 18t) + (4t^2 + 81t^2 + 9t^2)

D2=122+176t+94t2D^2 = 122 + 176t + 94t^2

To minimize D2D^2, take the derivative with respect to tt and set it to zero:

ddt(D2)=294t+176=188t+176\frac{d}{dt}(D^2) = 2 \cdot 94t + 176 = 188t + 176

Set the derivative equal to zero and solve for tt:

188t+176=0188t + 176 = 0

188t=176188t = -176

t=176188=8894=4447t = -\frac{176}{188} = -\frac{88}{94} = -\frac{44}{47}

Now substitute t=4447t = -\frac{44}{47} back into the parametric equations to find the coordinates (x,y,z)(x, y, z):

x=8+2(4447)=88847=376478847=28847x = 8 + 2\left(-\frac{44}{47}\right) = 8 - \frac{88}{47} = \frac{376}{47} - \frac{88}{47} = \frac{288}{47}

y=7+9(4447)=739647=3294739647=6747y = 7 + 9\left(-\frac{44}{47}\right) = 7 - \frac{396}{47} = \frac{329}{47} - \frac{396}{47} = -\frac{67}{47}

z=3+3(4447)=313247=1414713247=947z = 3 + 3\left(-\frac{44}{47}\right) = 3 - \frac{132}{47} = \frac{141}{47} - \frac{132}{47} = \frac{9}{47}

So, the coordinates of the point P(x,y,z)P(x, y, z) that is nearest to the origin are:

(28847,6747,947)\boxed{\left(\frac{288}{47}, -\frac{67}{47}, \frac{9}{47}\right)}

Would you like more details, or have any questions?


Here are some related questions you might find interesting:

  1. How do you derive the formula for the distance between a point and a line in 3D space?
  2. Can you find the distance from the origin to the line directly using a vector approach?
  3. How would the solution change if the line equation was given in a different form?
  4. What is the geometric interpretation of minimizing the squared distance in this context?
  5. Can you generalize this method to find the closest point on any line to any given point in space?

Tip: Remember that minimizing a function's square often simplifies the algebra, avoiding the complexities of square roots.

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Math Problem Analysis

Mathematical Concepts

Parametric equations
Distance minimization

Formulas

Distance formula in 3D

Theorems

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Suitable Grade Level

Advanced High School