Let's solve each question step-by-step, with explanations and final answers: --- ### 1. State whether each quantity is a vector or scalar. Explain. #### a. **Speed** - **Scalar**: Speed has magnitude only, no direction. #### b. **Velocity** - **Vector**: Velocity has both magnitude and direction. #### c. **Weight** - **Vector**: Weight is a force (mass × gravitational acceleration), has direction (towards Earth's center). #### d. **Mass** - **Scalar**: Mass has magnitude only, no direction. #### e. **Area** - **Scalar** (in basic context): Area usually refers to magnitude only. (In advanced physics, area can be treated as a vector with a normal direction.) --- ### 2. In the diagram, \( \triangle ACB \) is equilateral. Vertices **B**, **D**, and **P** are midpoints of \( AC, CE, \) and \( EA \). Let \( \overrightarrow{CB} = \vec{u} \) and \( \overrightarrow{EA} = \vec{v} \), \( \overrightarrow{AB} = \vec{u} \), \( \overrightarrow{AP} = \vec{v} \). Write the following in terms of \( \vec{u} \) and \( \vec{v} \): #### a. \( \overrightarrow{AC} \) - \( \overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} = \vec{u} + (-\vec{u}) = 0 \) But since \( AB = u \), and \( AC \) is a side, let's clarify: - \( \overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} = \vec{u} + (-\vec{v}) \) #### b. \( \overrightarrow{AD} \) - \( D \) is the midpoint of \( CE \). - \( \overrightarrow{AD} = \overrightarrow{AE} + \overrightarrow{ED} \) - \( \overrightarrow{AE} = -\overrightarrow{EA} = -\vec{v} \) - \( \overrightarrow{ED} = \frac{1}{2} \overrightarrow{EC} \) - \( \overrightarrow{EC} = \overrightarrow{EA} + \overrightarrow{AC} = \vec{v} + (\vec{u} + (-\vec{v})) = \vec{u} \) - So, \( \overrightarrow{ED} = \frac{1}{2} \vec{u} \) - Therefore, \( \overrightarrow{AD} = -\vec{v} + \frac{1}{2} \vec{u} \) #### c. \( \overrightarrow{CE} \) - \( \overrightarrow{CE} = \overrightarrow{CA} + \overrightarrow{AE} = -(\vec{u} + (-\vec{v})) + (-\vec{v}) = -\vec{u} + \vec{v} - \vec{v} = -\vec{u} \) - But \( CE \) is from \( C \) to \( E \), so: - \( \overrightarrow{CE} = \overrightarrow{CA} + \overrightarrow{AE} = -(\vec{u} + (-\vec{v})) + (-\vec{v}) = -\vec{u} + \vec{v} - \vec{v} = -\vec{u} \) #### d. \( \overrightarrow{BB} \) - This is a zero vector. \( \overrightarrow{BB} = \vec{0} \) --- #### Example of a vector equal to: ##### e. \( 2\vec{v} \) - \( \overrightarrow{AE} + \overrightarrow{EA} = \vec{v} + \vec{v} = 2\vec{v} \) ##### f. \( \vec{u} - \vec{v} \) - \( \overrightarrow{AB} + \overrightarrow{BC} = \vec{u} + (-\vec{v}) = \vec{u} - \vec{v} \) --- ### 3. Draw diagrams to show two vectors \( \vec{a} \) and \( \vec{b} \), and the vectors \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \). #### a. When is the magnitude of \( \vec{a} + \vec{b} \) more than that of \( \vec{a} - \vec{b} \)? - **When \( \vec{a} \) and \( \vec{b} \) point in the same direction.** #### b. When is the magnitude of \( \vec{a} + \vec{b} \) less than that of \( \vec{a} - \vec{b} \)? - **When \( \vec{a} \) and \( \vec{b} \) point in opposite directions.** #### c. When is the magnitude of \( \vec{a} + \vec{b} \) equal to that of \( \vec{a} - \vec{b} \)? - **When \( \vec{a} \) and \( \vec{b} \) are perpendicular.** #### d. When is \( |\vec{a} + \vec{b}| = |\vec{a}| + |\vec{b}| \)? - **When \( \vec{a} \) and \( \vec{b} \) are parallel and in the same direction.** #### e. When is the minimum value of \( |\vec{a} + \vec{b}| \)? - **When \( \vec{a} \) and \( \vec{b} \) are in opposite directions, minimum is \( ||\vec{a}| - |\vec{b}|| \).** --- ### 4. Boat sails 3 km South, then 10 km Southeast. Find the boat’s displacement and bearing. - First leg: 3 km South (\( 0^\circ \) or \( 270^\circ \) from East) - Second leg: 10 km Southeast (\( 135^\circ \) from East) #### Break into components: - South: \( (0, -3) \) - Southeast: \( 10 \) km at \( 45^\circ \) South of East (\( 135^\circ \) from East) - \( x = 10 \cos 45^\circ = 10 \times \frac{\sqrt{2}}{2} = 7.07 \) - \( y = -10 \sin 45^\circ = -7.07 \) Total: - \( x = 0 + 7.07 = 7.07 \) - \( y = -3 + (-7.07) = -10.07 \) Displacement: \( \sqrt{7.07^2 + 10.07^2} = \sqrt{49.98 + 101.41} = \sqrt{151.39} \approx 12.3 \) km Bearing: \( \theta = \tan^{-1}\left( \frac{7.07}{10.07} \right) = \tan^{-1}(0.702) \approx 35^\circ \) East of South **Final Answer:** **Displacement:** 12.3 km **Bearing:** \( 35^\circ \) East of South (or \( 180^\circ + 35^\circ = 215^\circ \) from East) --- ### 5. Convert the following vectors: #### a. 75 m/s on a bearing of 295° to Cartesian form 295° is 65° west of north, or 25° north of west. - \( x = 75 \cos 295^\circ = 75 \cos(-65^\circ) = 75 \times 0.4226 = 31.7 \) - \( y = 75 \sin 295^\circ = 75 \sin(-65^\circ) = 75 \times -0.9063 = -67.97 \) So, \( (x, y) = (31.7, -67.97) \) m/s #### b. \([-3, -8]\) to direction/magnitude form Magnitude: \( \sqrt{(-3)^2 + (-8)^2} = \sqrt{9 + 64} = \sqrt{73} \approx 8.54 \) Direction: \( \theta = \tan^{-1} \left( \frac{-8}{-3} \right) = \tan^{-1}(2.67) = 69^\circ \) (since both are negative, vector is in the third quadrant) --- ### 6. Express as a single vector: #### a. \( \vec{PS} + \vec{SR} \) - \( \vec{PS} + \vec{SR} = \vec{PR} \) #### b. \( \vec{FP} - \vec{DP} \) - \( \vec{FP} - \vec{DP} = \vec{FD} \) #### c. \( \vec{MP} - \vec{QR} + \vec{NM} - \vec{RD} \) - \( \vec{MP} + \vec{NM} = \vec{NP} \) - \( -\vec{QR} - \vec{RD} = -(\vec{QR} + \vec{RD}) = -\vec{QD} \) - So, sum is \( \vec{NP} - \vec{QD} \) --- ### 7. Demonstrate using diagrams: #### a. Commutative: \( \vec{a} + \vec{b} = \vec{b} + \vec{a} \) - **Explanation:** The order of addition does not affect the resultant vector. #### b. Associative: \( \vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c} \) - **Explanation:** The grouping of vectors does not affect the resultant vector. --- ### 8. Example of vectors in engineering, computer animation, gaming, 3D printing, or GPS: - **Example:** In GPS technology, vectors are used to calculate the shortest path (displacement) between two points on Earth's surface, and to give directions (bearing and distance) from the current location to the destination. This involves vector addition, subtraction, and conversion between coordinate systems. --- **If you need diagrams for any specific question, let me know!**