Question

The magnitude of projection of the line joining \( (3,4,5) \) and \( (4,6,3) \) on the line joining \( (-1,2,4) \) and \( (1,0,5) \) is:

• A. \( \frac{4}{3} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{8}{3} \)
• D. \( \frac{1}{3} \)

Hint:

Utilize the projection formula for vectors to find the component of one vector along another, simplifying the calculation of magnitudes in projections.

Answer 1

The Correct Option is A

The projection formula gives: \[ {Magnitude} = \frac{4}{3} \]

Answer 2

Step 1: Understand the Problem
We are asked to find the magnitude of the projection of the line joining the points \( A(3,4,5) \) and \( B(4,6,3) \) onto the line joining the points \( C(-1,2,4) \) and \( D(1,0,5) \).
This can be done by first calculating the direction vectors of the two lines and then using the formula for the projection of one vector onto another.
Step 2: Direction Vectors
The direction vector of the line \( AB \) is given by the difference of the coordinates of points \( B \) and \( A \):
\[ \vec{AB} = (4 - 3, 6 - 4, 3 - 5) = (1, 2, -2) \] Similarly, the direction vector of the line \( CD \) is given by the difference of the coordinates of points \( D \) and \( C \):
\[ \vec{CD} = (1 - (-1), 0 - 2, 5 - 4) = (2, -2, 1) \] Step 3: Projection Formula
The magnitude of the projection of vector \( \vec{AB} \) onto vector \( \vec{CD} \) is given by the formula:
\[ |\text{proj}_{\vec{CD}} \vec{AB}| = \frac{| \vec{AB} \cdot \vec{CD} |}{|\vec{CD}|} \] where \( \vec{AB} \cdot \vec{CD} \) is the dot product of \( \vec{AB} \) and \( \vec{CD} \), and \( |\vec{CD}| \) is the magnitude of \( \vec{CD} \).
Step 4: Calculate the Dot Product
The dot product of \( \vec{AB} = (1, 2, -2) \) and \( \vec{CD} = (2, -2, 1) \) is:
\[ \vec{AB} \cdot \vec{CD} = 1 \times 2 + 2 \times (-2) + (-2) \times 1 = 2 - 4 - 2 = -4 \] Step 5: Calculate the Magnitude of \( \vec{CD} \)
The magnitude of \( \vec{CD} = (2, -2, 1) \) is:
\[ |\vec{CD}| = \sqrt{2^2 + (-2)^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \] Step 6: Calculate the Projection
Now, we can calculate the magnitude of the projection:
\[ |\text{proj}_{\vec{CD}} \vec{AB}| = \frac{| -4 |}{3} = \frac{4}{3} \] Step 7: Conclusion
The magnitude of the projection of the line joining \( (3,4,5) \) and \( (4,6,3) \) onto the line joining \( (-1,2,4) \) and \( (1,0,5) \) is \( \frac{4}{3} \).