1In triangle ABC, which vector is equal to \( \overrightarrow{AB} + \overrightarrow{BC} \)? Choose from \( \overrightarrow{CA} \), \( \overrightarrow{BA} \), \( \overrightarrow{AC} \) and \( \overrightarrow{CB} \).
2In triangle ABC, D, E and F are the mid-points of the sides, with D on BC. Which of these is equal to \( 2\overrightarrow{AD} \): \( \overrightarrow{AB} - \overrightarrow{AC} \), \( \overrightarrow{AB} + \overrightarrow{AC} \), \( \overrightarrow{AC} - \overrightarrow{AB} \) or \( \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC}) \)?
3If the position vectors of A and B are \( \vec{a} \) and \( \vec{b} \), which is the position vector of the mid-point of AB: \( \vec{a} + \vec{b} \), \( \frac{\vec{a} - \vec{b}}{2} \), \( \frac{\vec{a} + \vec{b}}{2} \) or \( 2(\vec{a} + \vec{b}) \)?
4If \( \overrightarrow{OA} = 4\vec{i} + 3\vec{j} \) and \( \overrightarrow{OB} = 3\vec{i} - 4\vec{j} \), what is the position vector of the mid-point of AB? Choose from \( 7\vec{i} - \vec{j} \), \( \frac{1}{2}(7\vec{i} - \vec{j}) \), \( \vec{i} + 7\vec{j} \) and \( \frac{1}{2}(\vec{i} + 7\vec{j}) \).
5If P divides the line segment joining the points with position vectors \( \vec{a} \) and \( \vec{b} \) internally in the ratio m : n, which is the position vector \( \vec{p} \): \( \frac{m\vec{a} + n\vec{b}}{m+n} \), \( \frac{n\vec{a} + m\vec{b}}{m+n} \), \( \frac{n\vec{a} - m\vec{b}}{n-m} \) or \( \frac{m\vec{b} + n\vec{a}}{m-n} \)?
6The position vectors of P and Q are \( 4\vec{i} + 7\vec{j} \) and \( 4\vec{i} - 3\vec{j} \). Find the position vector of R, the mid-point of PQ.
7The position vectors of R and S are \( 5\vec{i} - 4\vec{j} \) and \( 3\vec{i} + 6\vec{j} \). Find the position vector of T, the mid-point of RS.
8A(2, \( -3 \)) and B(3, 2) are the ends of the line segment AB. M divides AB internally in the ratio 2 : 3. Find the position vector of M.
9A(2, \( -3 \)) and B(3, 2) are the ends of the line segment AB. N divides AB externally in the ratio 3 : 1. Find the position vector of N.
10The position vectors of A and B are \( \overrightarrow{OA} = 6\vec{i} - 2\vec{j} \) and \( \overrightarrow{OB} = 2\vec{i} + 3\vec{j} \). P divides AB internally in the ratio 3 : 1. Find the position vector of P.
11The position vectors of A and B are \( \overrightarrow{OA} = 6\vec{i} - 2\vec{j} \) and \( \overrightarrow{OB} = 2\vec{i} + 3\vec{j} \). Q divides AB externally in the ratio 2 : 1. Find the position vector of Q.
12In triangle ABC, F and E are the mid-points of AB and AC. Write the statement of the mid-point theorem in vectors.
13In triangle ABC, use the triangle law of addition to express \( \overrightarrow{BC} \) in terms of \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \).
14In triangle ABC, F and E are the mid-points of AB and AC. Prove by the vector method that \( \overrightarrow{EF} = \frac{1}{2}\overrightarrow{CB} \), or equivalently \( \overrightarrow{FE} = \frac{1}{2}\overrightarrow{BC} \).
15In triangle ABC, D, E and F are the mid-points of BC, CA and AB. Prove that \( \overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} = \vec{0} \).
16The position vectors of A and B are \( \vec{a} \) and \( \vec{b} \), and P divides AB internally in the ratio m : n. Write the formula for the position vector \( \vec{p} \) of P.
17P divides AB internally so that AP : PB = m : n. Why is this written in vectors as \( n\overrightarrow{AP} = m\overrightarrow{PB} \)? Give a reason.
18If \( \vec{a} = \vec{i} + 2\vec{j} \) and \( \vec{b} = 4\vec{i} + 8\vec{j} \), find the position vector of the point that divides AB internally in the ratio 1 : 2.
19In which case is \( \vec{p} = \frac{\vec{a} + \vec{b}}{2} \)? Give a reason.
20P divides the line segment joining \( \vec{a} \) and \( \vec{b} \) externally in the ratio m : n. Write the formula for the position vector \( \vec{p} \).
21Prove by the vector method that \( \vec{p} = \frac{m\vec{b} - n\vec{a}}{m-n} \) for a point P dividing AB externally in the ratio m : n.
22If \( \vec{a} = \vec{i} + 4\vec{j} \) and \( \vec{b} = 5\vec{i} - 3\vec{j} \), find the position vector of the point that divides AB externally in the ratio 1 : 2.
23If m : n = 1 : 1, does a point P dividing AB externally exist? Explain with reasons.
24Prove by the vector method that the quadrilateral formed by joining the mid-points of the sides of a quadrilateral in order is a parallelogram.
25Prove by the vector method that the angle in a semicircle is a right angle.
26Two vectors are \( \vec{a} = 3\vec{i} + 4\vec{j} \) and \( \vec{b} = x\vec{i} - 2\vec{j} \). If \( \vec{a} = (x_1, y_1) \) and \( \vec{b} = (x_2, y_2) \), write \( \vec{a}\cdot\vec{b} \) in terms of the components.
27For \( \vec{a} = 3\vec{i} + 4\vec{j} \) and \( \vec{b} = x\vec{i} - 2\vec{j} \), find x if \( \vec{a} \) and \( \vec{b} \) are perpendicular.
28For \( \vec{b} = x\vec{i} - 2\vec{j} \), find the value of x for which \( \vec{b} \) makes an angle of \( 45^\circ \) with the Y-axis.
29The position vectors of A and B are \( 2\vec{i} + 4\vec{j} \) and \( 4\vec{i} - 2\vec{j} \). Find the position vector of the mid-point of AB.
30In triangle ABC, \( \overrightarrow{OA} = 2\vec{i} + 3\vec{j} \), \( \overrightarrow{OB} = 4\vec{i} + \vec{j} \) and \( \overrightarrow{OC} = 6\vec{i} + 5\vec{j} \). If D is the mid-point of AB, express \( \overrightarrow{OD} \) in terms of \( \vec{i} \) and \( \vec{j} \).
31In the same triangle, if E is the mid-point of AC, express \( \overrightarrow{OE} \) in terms of \( \vec{i} \) and \( \vec{j} \).
32In the same triangle, prove by the vector method that \( \overrightarrow{DE} = \frac{1}{2}\overrightarrow{BC} \).
33In the same triangle, check whether \( \overrightarrow{DE} \) and \( \overrightarrow{BC} \) are parallel.
34The position vectors of P and Q are \( \vec{p} = 2\vec{i} + \vec{j} \) and \( \vec{q} = \vec{i} - 3\vec{j} \). R divides PQ in the ratio 3 : 2. Write the formula for the position vector of a point dividing the line joining two points in the ratio m : n.
35With \( \vec{p} = 2\vec{i} + \vec{j} \) and \( \vec{q} = \vec{i} - 3\vec{j} \), find the position vector \( \vec{r} \) of R, which divides PQ in the ratio 3 : 2.
36With P, Q and R as above, find \( \angle POR \).
37With P, Q and R as above, compare \( \angle POR \) and \( \angle QOR \).
38The position vectors of A and B are \( 3\vec{i} + 2\vec{j} \) and \( \vec{i} - 4\vec{j} \). P divides AB externally in the ratio 2 : 3. Write the formula for the position vector of a point dividing the line joining two points externally in the ratio m : n.
39With A and B as above, find the position vector of P, which divides AB externally in the ratio 2 : 3.
40If m > n, is a point P dividing AB externally nearer to A or to B? Give a reason.
41With \( \overrightarrow{OA} = 3\vec{i} + 2\vec{j} \) and \( \overrightarrow{OB} = \vec{i} - 4\vec{j} \), find \( \angle AOB \).
42In the proof that the mid-point of the hypotenuse is equidistant from the vertices, why may we write \( \overrightarrow{DC} = -\overrightarrow{DA} \)?
43Why does the proof that the angle in a semicircle is a right angle end with a scalar product rather than with a length?
44In the proof that the mid-points of a quadrilateral form a parallelogram, what construction is made and why?
45The position vectors of A and B are \( 2\vec{i} + 6\vec{j} \) and \( 8\vec{i} - 4\vec{j} \). Find the position vector of the point M dividing AB internally in the ratio 1 : 1, and say what M is.