BITSAT 2024 Shift 1 Paper

The projection of a vector a on another vector b is calculated using the formula:
|a|‌cos‌θ=‌

(a⋅b)

|b|

Consider the line joining the points (3,4,5) and (4,6,3) as L1, and the line joining the points (−1,2,4) and (1,0,5) as L2.
The direction vector L1 is derived as follows:
L1=(4−3)

∧

i

+(6−4)

∧

j

+(3−5)

∧

k

=

∧

i

+2

∧

j

−2

∧

k

Similarly, the direction vector L2 is:
L2=(1−(−1))

∧

i

+(0−2)

∧

j

+(5−4)

∧

k

=2

∧

i

−2

∧

j

+

∧

k

The projection of L1 onto L2 is calculated by:
Projection of L1 on L2=‌

L1⋅L2

|L2|

Calculating the dot product L1⋅L2 :
L1⋅L2=(1×2)+(2×−2)+(−2×1)=2−4−2=−4
The magnitude of L2 is:
|L2|=√(2)2+(−2)2+(1)2=√4+4+1=√9=3
Thus, the projection is:
‌

−4

3

Therefore, the magnitude of the projection is:
‌

4

3