The helix $\mathbf { r } _ { 1 } ( t ) = 4 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ intersects the curve $\mathbf { r } _ { 2 } ( t ) = ( 4 + t ) \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k }$ at the point $( 4,0,0 )$. Find the angle of intersection.
A) 0
B) $\frac { \pi } { 4 }$
C) $\frac { \pi } { 2 }$
D) $\frac { \pi } { 3 }$
E) None of these

Question

Quizplus · Accepted Answer

The angle of intersection is determined by the dot product of the tangent vectors at the point of intersection. The tangent vectors are perpendicular when their dot product is zero, indicating an angle of $\frac{\pi}{2}$.

The Helix r1(t)=4cos⁡ti+sin⁡tj+tk\mathbf { r } _ { 1 } ( t ) = 4 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }r1​(t)=4costi+sintj+tk

The Helix $\mathbf { r } _ { 1 } ( t ) = 4 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$