Introduction
A vector-valued function assigns a vector to each value of a variable such as time, so it can describe the position of a moving object. Unit 11 of the PECTAA New Book covers constructing and classifying vector functions, their domain and range, their derivatives, and the velocity and acceleration of a particle in a plane or in space.
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What This Unit Covers
Unit 11 is organized into three graded exercises (11.1 to 11.3):
1. Scalar and Vector-Valued Functions (Exercise 11.1) Students distinguish scalar functions from vector functions, construct vector functions for real motion such as a circular path, a parabolic path or a spiral, and find the domain and range of a given function.
2. Derivatives of Vector-Valued Functions (Exercise 11.2) Students differentiate a vector function component by component, evaluate derivatives at given values of t, and link the derivative to velocity and the second derivative to acceleration.
3. Velocity, Acceleration and Motion Analysis (Exercise 11.3) Students find velocity, acceleration, speed and direction of motion at a given time, interpret whether the particle is speeding up or slowing down, and solve problems on cannonballs, comets, robotic arms and circular motion.
Why This Unit Matters
Vector-valued functions unite calculus with vectors and are used in physics, engineering and computer graphics. They bring together derivative rules from Unit 2 and the kinematics ideas of Unit 7.
How to Study Unit 11
- Write the function as components and differentiate each component with the correct rule.
- Remember: velocity is r′(t), acceleration is r″(t), and speed is the magnitude of the velocity.
- State the domain by combining the domains of all components.
- For projectile problems, write the horizontal and vertical components separately.
- To show two vectors are perpendicular, show their dot product is zero.
Practice Questions – Unit 11
(Original practice problems for self-assessment. Attempt them after studying the unit, then check your working against the solved examples in your notes.)
Short Questions
- Define a vector-valued function.
- Find the domain of r(t) = √t î + ln t ĵ.
- Find r′(t) if r(t) = t î + t² ĵ.
- How are velocity and acceleration obtained from the position vector r(t)?
- Find the speed of a particle whose velocity is 3î + 4ĵ.
- Is f(t) = t² + sin t a scalar or a vector function?
Long Questions
- Construct a vector function for a particle moving on a circle of radius 3.
- Find r′(t) and r″(t) for r(t) = (t² + 1) î + e^(2t) ĵ + sin t k̂.
- For r(t) = cos t î + sin t ĵ + t k̂, find the velocity, speed and acceleration at t = π/2.
- Show that for r(t) = R cos ωt î + R sin ωt ĵ the velocity is perpendicular to the position vector.
- Find r′(1) for r(t) = (t ln t) î + (e^(−t) sin t) ĵ.
Applied Questions
- A ball is thrown at 20 m/s at 30° above the horizontal. Taking g = 10 m/s², write its position vector and find the time of flight.
- A drone follows r(t) = 2t î + t² ĵ + 3t k̂. Find its velocity and speed at t = 2.
- A satellite moves so that r(t) = 7000 cos t î + 7000 sin t ĵ (km). Find its speed and show that its acceleration has constant magnitude.
- A robotic arm has x = 3t, y = t² and z = 1 − e^(−t). Construct r(t) and find its velocity at t = 1.
These practice questions are written to reinforce Unit 11 concepts and are separate from the textbook’s own exercises. Use them for extra revision before attempting the board-style exercises in the unit.
About HSA Notes
HSA Notes is an educational resource platform providing free study materials for Pakistani students, covering FBISE, Punjab, Sindh, and KP Boards in line with the New Books System. Materials span grades 9–12 across Mathematics, Chemistry, Biology, Physics, Computer Science, English, and Urdu, all prepared in a consistent, easy-to-follow branded format.
Besides free unit-wise notes and solved exercises, HSA Notes also offers online classes for students who want structured, teacher-guided sessions alongside self-study, useful for learners who need concepts explained step by step or want doubt-solving support beyond what a PDF can provide. For online classes, contact: hsanotes48@gmail.com
Conclusion
Unit 11 – Vector Valued Functions and Their Derivatives extends calculus to motion in two and three dimensions. Differentiating component by component and interpreting the results as velocity, speed and acceleration gives students a compact, powerful method. Practice with the questions above, together with HSA Notes’ online classes where needed, will build confidence.
