Vector Operations, Differentiation, and Motion in Physics: Step-by-Step Guidance

Q1(a). Find the magnitude and direction (including appropriate signs) of \( \vec{A} = 3\hat{i} - \hat{j} \) and \( \vec{B} = \hat{i} + \hat{j} \).

Background

Topic: Vectors in Physics

This question tests your understanding of vector representation, magnitude, and direction in two dimensions.

Key Terms and Formulas

• Magnitude of a vector \( \vec{V} = a\hat{i} + b\hat{j} \): $|\vec{V}| = \sqrt{a^2 + b^2}$

• Direction (angle \( \theta \) with respect to the x-axis): $\theta = \arctan\left(\frac{b}{a}\right)$

Step-by-Step Guidance

1. Identify the components of each vector: \( \vec{A} = 3\hat{i} - \hat{j} \) has components (3, -1); \( \vec{B} = 1\hat{i} + 1\hat{j} \) has components (1, 1).

2. Calculate the magnitude of each vector using $|\vec{V}| = \sqrt{a^2 + b^2}$.

3. Find the direction (angle from the x-axis) for each vector using $\theta = \arctan\left(\frac{b}{a}\right)$. Pay attention to the signs of the components to determine the correct quadrant.

Try solving on your own before revealing the answer!

Q1(b). Plot each vector on an x-y axis with their tails at the origin. The tip of each vector should be located carefully with spaced markings on the x– and y–axes.

Background

Topic: Vector Representation in Cartesian Coordinates

This part checks your ability to graphically represent vectors using their components.

Key Terms and Formulas

• Vector components: The x-component is the coefficient of \( \hat{i} \), and the y-component is the coefficient of \( \hat{j} \).

Step-by-Step Guidance

1. On a set of x-y axes, mark the origin (0,0).

2. For \( \vec{A} \), start at the origin and move 3 units along the x-axis and -1 unit along the y-axis to locate the tip.

3. For \( \vec{B} \), start at the origin and move 1 unit along the x-axis and 1 unit along the y-axis to locate the tip.

4. Draw arrows from the origin to each tip, labeling them as \( \vec{A} \) and \( \vec{B} \).

Try sketching the vectors before checking the answer!

Q1(c). Use the dot product to find the angle between \( \vec{A} \) and \( \vec{B} \).

Background

Topic: Dot Product and Angle Between Vectors

This question tests your ability to use the dot product to determine the angle between two vectors.

Key Terms and Formulas

• Dot product: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y$

• Relation to angle: $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$

• Solving for angle: $\theta = \arccos\left(\frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}\right)$

Step-by-Step Guidance

1. Calculate the dot product using the components of \( \vec{A} \) and \( \vec{B} \).

2. Find the magnitudes of \( \vec{A} \) and \( \vec{B} \) (from part a).

3. Plug the values into $\theta = \arccos\left(\frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}\right)$ to set up the calculation for the angle.

Try setting up the calculation before revealing the answer!

Q1(d). Let \( \vec{C} = \vec{A} + \vec{B} \) and \( \vec{D} = \vec{A} - \vec{B} \). Write \( \vec{C} \) and \( \vec{D} \) in component form and add each one to your drawing above in the same manner as \( \vec{A} \) and \( \vec{B} \).

Background

Topic: Vector Addition and Subtraction

This part tests your ability to add and subtract vectors using their components and represent the results graphically.

Key Terms and Formulas

• Vector addition: $\vec{C} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$

• Vector subtraction: $\vec{D} = \vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j}$

Step-by-Step Guidance

1. Add the corresponding components of \( \vec{A} \) and \( \vec{B} \) to find \( \vec{C} \).

2. Subtract the components of \( \vec{B} \) from \( \vec{A} \) to find \( \vec{D} \).

3. Write both \( \vec{C} \) and \( \vec{D} \) in component form (\( \hat{i}, \hat{j} \)).

4. On your previous plot, add arrows for \( \vec{C} \) and \( \vec{D} \) starting from the origin, using their new components.

Try writing the component forms before checking the answer!

Q1(e). Let \( \vec{F} = \vec{A} \times \vec{B} \). Find \( \vec{F} \) in component form using the determinant and add it to your drawing above.

Background

Topic: Cross Product of Vectors

This question tests your ability to compute the cross product using the determinant method.

Key Terms and Formulas

• Cross product (determinant form): $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$

• For vectors in the x-y plane, $A_z = B_z = 0$.

Step-by-Step Guidance

1. Write the components of \( \vec{A} \) and \( \vec{B} \), including zeros for the z-components.

2. Set up the determinant for the cross product.

3. Expand the determinant to find the components of \( \vec{F} \).

Try expanding the determinant before revealing the answer!

Q1(f). Now find \( \vec{F} \) using the right-hand rule and \( |\vec{A} \times \vec{B}| = AB\sin\theta \). Show that it agrees with what you found in (e).

Background

Topic: Magnitude and Direction of Cross Product

This part checks your understanding of the geometric interpretation of the cross product and the right-hand rule.

Key Terms and Formulas

• Magnitude: $|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta$

• Direction: Use the right-hand rule to determine the direction (along the z-axis for vectors in the x-y plane).

Step-by-Step Guidance

1. Recall the angle \( \theta \) between \( \vec{A} \) and \( \vec{B} \) from part (c).

2. Calculate the magnitudes of \( \vec{A} \) and \( \vec{B} \) (from part a).

3. Set up $|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta$ using your previous results.

4. Use the right-hand rule to determine the direction (positive or negative z-axis).

Try relating the two methods before checking the answer!

Q1(g). Find a unit vector that points in the same direction as \( \vec{B} \).

Background

Topic: Unit Vectors

This question tests your ability to find a unit vector in the direction of a given vector.

Key Terms and Formulas

• Unit vector: $\hat{u}_B = \frac{\vec{B}}{|\vec{B}|}$

Step-by-Step Guidance

1. Find the magnitude of \( \vec{B} \) (from part a).

2. Divide each component of \( \vec{B} \) by its magnitude to get the unit vector.

Try writing the unit vector before revealing the answer!

Q2(a). Differentiate \( g(x) = (x^2 + 1) \sin x \) with respect to x.

Background

Topic: Differentiation (Product Rule)

This question tests your ability to use the product rule for differentiation.

Key Terms and Formulas

• Product rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

• Derivative of $\sin x$: $\frac{d}{dx}(\sin x) = \cos x$

Step-by-Step Guidance

1. Let $u(x) = x^2 + 1$ and $v(x) = \sin x$.

2. Find $u'(x)$ and $v'(x)$.

3. Apply the product rule formula to set up the derivative.

Try applying the product rule before revealing the answer!

Q2(b). Differentiate \( h(x) = \frac{x^2 + 3x + 1}{x - 1} \) with respect to x.

Background

Topic: Differentiation (Quotient Rule)

This question tests your ability to use the quotient rule for differentiation.

Key Terms and Formulas

• Quotient rule: $\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

Step-by-Step Guidance

1. Let $u(x) = x^2 + 3x + 1$ and $v(x) = x - 1$.

2. Find $u'(x)$ and $v'(x)$.

3. Apply the quotient rule formula to set up the derivative.

Try applying the quotient rule before revealing the answer!

Q2(c). Differentiate \( q(x) = \cos(4x) \) with respect to x.

Background

Topic: Differentiation (Chain Rule)

This question tests your ability to use the chain rule for differentiation.

Key Terms and Formulas

• Chain rule: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

• Derivative of $\cos u$: $\frac{d}{du}(\cos u) = -\sin u$

Step-by-Step Guidance

1. Let $u = 4x$, so $q(x) = \cos(u)$.

2. Find $\frac{d}{dx}(\cos(u))$ using the chain rule.

3. Multiply by the derivative of the inner function $u$ with respect to $x$.

Try applying the chain rule before revealing the answer!

Q3(a). A particle in space has position vector \( \vec{r}(t) = t^2 \hat{i} + (3t - 1) \hat{j} + (2 - t^3) \hat{k} \). Find the velocity vector \( \vec{v}(t) = \frac{d\vec{r}}{dt} \).

Background

Topic: Kinematics in Three Dimensions

This question tests your ability to find the velocity vector by differentiating the position vector with respect to time.

Key Terms and Formulas

• Velocity vector: $\vec{v}(t) = \frac{d\vec{r}}{dt}$

Step-by-Step Guidance

1. Differentiate each component of \( \vec{r}(t) \) with respect to $t$ separately.

2. Write the resulting velocity vector in component form: $v_x(t)\hat{i} + v_y(t)\hat{j} + v_z(t)\hat{k}$.

Try differentiating each component before revealing the answer!

Q3(b). Find the speed $|\vec{v}(t)|$.

Background

Topic: Magnitude of Velocity (Speed)

This question tests your ability to find the magnitude of a vector function.

Key Terms and Formulas

• Speed: $|\vec{v}(t)| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

Step-by-Step Guidance

1. Use the velocity components from part (a).

2. Square each component, sum them, and take the square root to set up the expression for speed.

Try setting up the magnitude before revealing the answer!

Q3(c). Find the acceleration vector $\vec{a}(t) = \frac{d\vec{v}}{dt}$.

Background

Topic: Acceleration in Three Dimensions

This question tests your ability to find the acceleration vector by differentiating the velocity vector with respect to time.

Key Terms and Formulas

• Acceleration vector: $\vec{a}(t) = \frac{d\vec{v}}{dt}$

Step-by-Step Guidance

1. Differentiate each component of the velocity vector (from part a) with respect to $t$.

2. Write the resulting acceleration vector in component form: $a_x(t)\hat{i} + a_y(t)\hat{j} + a_z(t)\hat{k}$.

Try differentiating each component before revealing the answer!

Q4(a). Let $f(x) = x^3 - 6x^2 + 9x$. Find all critical points (minima and maxima) of $f$.

Background

Topic: Critical Points of a Function

This question tests your ability to find critical points by setting the first derivative to zero.

Key Terms and Formulas

• Critical points: Values of $x$ where $f'(x) = 0$ or $f'(x)$ is undefined.

• First derivative: $f'(x)$

Step-by-Step Guidance

1. Find the first derivative $f'(x)$.

2. Set $f'(x) = 0$ and solve for $x$ to find the critical points.

Try finding the derivative and setting it to zero before revealing the answer!

Q4(b). Determine which point from part (a) is a minimum and which is a maximum.

Background

Topic: Second Derivative Test

This question tests your ability to classify critical points using the second derivative test.

Key Terms and Formulas

• Second derivative: $f''(x)$

• If $f''(x) > 0$ at a critical point, it's a minimum; if $f''(x) < 0$, it's a maximum.

Step-by-Step Guidance

1. Find the second derivative $f''(x)$.

2. Plug each critical point into $f''(x)$ to determine if it's a minimum or maximum.

Try applying the second derivative test before revealing the answer!