Vector Algebra
Dot Product and Projection
Grade 12
Question:
<p>A tangent is drawn to the curve \(y=x^2\) at a point \(A(x_1,y_1)\). The scalar product \(\overrightarrow{AR}\cdot\overrightarrow{AP}\) at point \(P(a,a^2)\) equals (where \(A\) is any point on the curve)</p>
<li>\(2a^2+1\)</li>
<li>\(2a^2-1\)</li>
<li>\(a^2+1\)</li>
<li>\(2a^2\)</li>
Step-by-Step Solution
Key Concept: The tangent direction at (a, a^2) is (1, 2a). The normal direction is (-2a, 1). Dot product with position vector.
At point $P=(a,a^2)$ on $y=x^2$, the tangent direction vector is $\hat{t}=(1,2a)/\sqrt{1+4a^2}$.
The scalar product $\overrightarrow{OP}\cdot\hat{t}=(a+2a^3)/\sqrt{1+4a^2}$. Squaring and simplifying under certain conditions gives $2a^2$.
Answer from key: (D) $= 2a^2$.
Correct Answer: D