Curtin University Extra Exercise PDF
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Curtin University
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This document contains extra exercises in vector calculus, covering topics such as calculating magnitudes and directions of vectors, vector equations of lines, and scalar product of vectors, suitable for undergraduate-level mathematics students at Curtin University.
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# Curtin University Malaysia ## Extra Exercise 1. Given that **a** = 5 + 5**j** and **b** = 2 − 8**j**. Calculate the magnitude and direction of - **a** - **b** 2. OABC is a trapezium with **OA** = **a** and **OB** = **b**. OA is parallel to CB, and twice as long as CB. Express each of the foll...
# Curtin University Malaysia ## Extra Exercise 1. Given that **a** = 5 + 5**j** and **b** = 2 − 8**j**. Calculate the magnitude and direction of - **a** - **b** 2. OABC is a trapezium with **OA** = **a** and **OB** = **b**. OA is parallel to CB, and twice as long as CB. Express each of the following in terms of **a** or **b**. - **CB** - **BA** - **CA** - **CO** 3. Given that **a** = 2**i** + **j** and **c** = 2**i** + 3**j**, **b** = 2**i** + 3**j** and **c** = 2**i** + 3**j**. Calculate the values of the **k** and **m** if - **c** = **k** **a** + **m** **b** 4. Given that **a** = (<sup>1</sup>/<sub>2</sub>)**i** + (<sup>1</sup>/<sub>2</sub>)**j**, **b** = (<sup>1</sup>/<sub>3</sub>)**i** + (<sup>1</sup>/<sub>3</sub>)**j** and **c** = (<sup>1</sup>/<sub>5</sub>)**i** + (<sup>1</sup>/<sub>5</sub>)**j**. Calculate the - values of **x** and **p** if **b**+ **p** **c** = (<sup>-10</sup>/<sub>3</sub>)**j**. - value of **y** if **b** is parallel to **c**. - unit vector which is parallel to **a** + **b** + **c** using the values of **x** and **y** found in (a) and (b). 5 Calculate a vector of magnitude 14 in the direction of the vector, **v** = 6**i** - 3**j** + 2**k**. 6. Given that **a** = 2**i** + 5**j** and **b** = 3**i** − 2**j**. Calculate - **a** + **b** - 2**a** − 3**b** 7. If **PQ** = (<sup>2</sup>/<sub>2</sub> , <sup>4</sup>/<sub>2</sub>) and **PR** = (<sup>2</sup>/<sub>4</sub>, <sup>-4</sup>/<sub>1</sub>). Calculate the size of angle QPR. Give your angle in degrees accurate to two decimal places. # Extra Exercise - Vector Equation of a Line 1. Develop a vector equation of a line which passes through (3, 2) and then (2, -1). Hence identify whether the point (2, -5) lies on the line. 2. Given that **AB** = 10**i** + 12**j** and **BC** = 4**i** - 5**j**. Evaluate **AC** and hence determine the distance between A and C. # Extra Exercise (Answers) 1. √388, 66.04° 2. - (<sup>1</sup>/<sub>3</sub>) **a** + **b** - (<sup>1</sup>/<sub>3</sub>) **a** - **b** - (<sup>1</sup>/<sub>2</sub>) **a** - **b** - (<sup>1</sup>/<sub>2</sub>) **a** - **b** 3. m = <sup>-3</sup>/<sub>8</sub>, k = <sup>15</sup>/<sub>8</sub> 4. - x = -22, p = 1 - b = -4 - (<sup>4</sup>/<sub>5</sub>, <sup>-5</sup>/<sub>5</sub>) 5. 12**i** − 6**j** + 4**k** 6. - 5**i** + 3**j** - -5**i** + 19**j** 7. 128.25º 8. - 99.5º - 127.9º 9. - (<sup>5</sup>/<sub>32</sub>) - (<sup>3</sup>/<sub>5</sub>) # Extra Exercise - Vector Equation of a Line 1. **r** = ((2-3**t**) , (**-1**-3**t**)), t = scalar or **r** = ((**-1**-3**μ**), (**-1**-3**μ**)), µ = scalar; No 2. (<sup>14</sup>/<sub>2</sub>, <sup>7</sup>/<sub>5</sub>), 7√5
