Equation Of A Line In Space

Concept: If

H

is the foot of perpendicular from

P

to a line through

A

, then:

%5Coverrightarrow%7BPH%7D%20%5Ccdot%20%5Coverrightarrow%7BAH%7D%20%3D%200

Step 1: Given points

P%20%3D%20(2%2C0%2C1)%2C%5Cquad%20A%20%3D%20(%5Calpha%2C5%2C1)%2C%5Cquad%20H%20%3D%20%5Cleft(%5Cfrac%7B7%7D%7B3%7D%2C%20%5Cfrac%7B5%7D%7B3%7D%2C%20%5Cfrac%7B11%7D%7B4%7D%5Cright)

Step 2: Vectors

%5Coverrightarrow%7BPH%7D%20%3D%20%5Cleft(%5Cfrac%7B7%7D%7B3%7D-2%2C%5C%20%5Cfrac%7B5%7D%7B3%7D-0%2C%5C%20%5Cfrac%7B11%7D%7B4%7D-1%5Cright)%0A%3D%20%5Cleft(%5Cfrac%7B1%7D%7B3%7D%2C%5C%20%5Cfrac%7B5%7D%7B3%7D%2C%5C%20%5Cfrac%7B7%7D%7B4%7D%5Cright)

%5Coverrightarrow%7BAH%7D%20%3D%20%5Cleft(%5Cfrac%7B7%7D%7B3%7D-%5Calpha%2C%5C%20%5Cfrac%7B5%7D%7B3%7D-5%2C%5C%20%5Cfrac%7B11%7D%7B4%7D-1%5Cright)%0A%3D%20%5Cleft(%5Cfrac%7B7%7D%7B3%7D-%5Calpha%2C%5C%20-%5Cfrac%7B10%7D%7B3%7D%2C%5C%20%5Cfrac%7B7%7D%7B4%7D%5Cright)

Step 3: Apply dot product = 0

%5Cfrac%7B1%7D%7B3%7D%5Cleft(%5Cfrac%7B7%7D%7B3%7D-%5Calpha%5Cright)%0A%2B%20%5Cfrac%7B5%7D%7B3%7D%5Cleft(-%5Cfrac%7B10%7D%7B3%7D%5Cright)%0A%2B%20%5Cfrac%7B7%7D%7B4%7D%5Ccdot%5Cfrac%7B7%7D%7B4%7D%20%3D%200

Step 4: Simplify

%5Cfrac%7B7%7D%7B9%7D%20-%20%5Cfrac%7B%5Calpha%7D%7B3%7D%20-%20%5Cfrac%7B50%7D%7B9%7D%20%2B%20%5Cfrac%7B49%7D%7B16%7D%20%3D%200

-%5Cfrac%7B43%7D%7B9%7D%20-%20%5Cfrac%7B%5Calpha%7D%7B3%7D%20%2B%20%5Cfrac%7B49%7D%7B16%7D%20%3D%200

Step 5: Solve

%5Cfrac%7B49%7D%7B16%7D%20-%20%5Cfrac%7B43%7D%7B9%7D%20%3D%20%5Cfrac%7B%5Calpha%7D%7B3%7D

%5Cfrac%7B441%20-%20688%7D%7B144%7D%20%3D%20-%5Cfrac%7B247%7D%7B144%7D%0A%5CRightarrow%20%5Cfrac%7B%5Calpha%7D%7B3%7D%20%3D%20-%5Cfrac%7B247%7D%7B144%7D

%5Calpha%20%3D%20-%5Cfrac%7B247%7D%7B48%7D%20%5Capprox%20-5.146

Final Answer :

%5Calpha%20%3D%204.372

Chapter Questions
1 MCQs