• Syllabus

4.1

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Sub sections and their related questions

Concept of a vector.

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 11M.2.hl.TZ2.11a: Find the coordinates of S.
• SPNone.2.hl.TZ0.8a: Show that OC = 2AB .
• SPNone.2.hl.TZ0.8b: Find the position vectors of C, D and E in terms of a and b .
• 13M.1.hl.TZ2.11a: (i)     Find the lengths of the sides of the triangle. (ii)     Find \(\cos {\rm{B\hat AC}}\).
• 11M.1.hl.TZ1.4a: Write down expressions for \(\overrightarrow {{\text{AB}}} \) and...
• 11M.1.hl.TZ1.11a: Find the vectors \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{AC}}} \).
• 18M.2.hl.TZ1.11a: Show that the two submarines would collide at a point P and write down the coordinates of P.

Representation of vectors using directed line segments.

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 18M.2.hl.TZ1.11a: Show that the two submarines would collide at a point P and write down the coordinates of P.

Unit vectors; base vectors \(i\), \(j\), \(k\).

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 18M.2.hl.TZ1.11a: Show that the two submarines would collide at a point P and write down the coordinates of P.

Components of a vector: \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 18M.2.hl.TZ1.11a: Show that the two submarines would collide at a point P and write down the coordinates of P.

Algebraic and geometric approaches to the sum and difference of two vectors.

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 10M.1.hl.TZ2.3: The three vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\) and \(\boldsymbol{c}\) are given...
• 10N.1.hl.TZ0.10: Let \(\alpha \) be the angle between the unit vectors a and b, where...
• 14M.1.hl.TZ1.12b: Find the coordinates of M, the mid-point of [OB].
• 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}}  = \) \(\boldsymbol{a}\) and...
• 14M.1.hl.TZ1.12a: Show that the points \({\text{O}}(0,{\text{ }}0,{\text{ }}0)\),...
• 15N.1.hl.TZ0.13a: Find \(\overrightarrow {{\text{BR}}} \) in terms of \({{a}}\), \({{b}}\) and \({{c}}\).
• 15N.2.hl.TZ0.10a: find the vectors \({{a}}\) and \({{b}}\).
• 16M.1.hl.TZ1.8: O, A, B and C are distinct points such that \(\overrightarrow {{\text{OA}}}  = \) a,...
• 16M.2.hl.TZ2.9a: Show that (i)     \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)a\({|^2} +...
• 16M.2.hl.TZ2.9b: Given that...
• 17N.1.hl.TZ0.9a.i: Find, in terms of a and b \(\overrightarrow {{\text{OF}}} \).
• 17N.1.hl.TZ0.9a.ii: Find, in terms of a and b \(\overrightarrow {{\text{AF}}} \).
• 17N.1.hl.TZ0.9b.i: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\lambda \);
• 17N.1.hl.TZ0.9b.ii: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\mu \).
• 17N.1.hl.TZ0.9c: Show that \(\mu  = \frac{1}{{13}}\), and find the value of \(\lambda \).
• 17N.1.hl.TZ0.9d: Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of a and b only.
• 17N.1.hl.TZ0.9e: Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of...
• 18M.1.hl.TZ2.9a.i: Explain why ABCD is a parallelogram.
• 18M.1.hl.TZ2.9a.ii: Using vector algebra, show that...
• 18M.1.hl.TZ2.9b: Show that p = 1, q = 1 and r = 4.
• 18M.1.hl.TZ2.9c: Find the area of the parallelogram ABCD.

Algebraic and geometric approaches to the zero vector \(0\), the vector \( - v\) .

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 14M.1.hl.TZ1.12b: Find the coordinates of M, the mid-point of [OB].
• 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}}  = \) \(\boldsymbol{a}\) and...
• 14M.1.hl.TZ1.12a: Show that the points \({\text{O}}(0,{\text{ }}0,{\text{ }}0)\),...
• 16M.1.hl.TZ1.8: O, A, B and C are distinct points such that \(\overrightarrow {{\text{OA}}}  = \) a,...
• 16M.2.hl.TZ2.9a: Show that (i)     \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)a\({|^2} +...
• 16M.2.hl.TZ2.9b: Given that...
• 17N.1.hl.TZ0.9a.i: Find, in terms of a and b \(\overrightarrow {{\text{OF}}} \).
• 17N.1.hl.TZ0.9a.ii: Find, in terms of a and b \(\overrightarrow {{\text{AF}}} \).
• 17N.1.hl.TZ0.9b.i: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\lambda \);
• 17N.1.hl.TZ0.9b.ii: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\mu \).
• 17N.1.hl.TZ0.9c: Show that \(\mu  = \frac{1}{{13}}\), and find the value of \(\lambda \).
• 17N.1.hl.TZ0.9d: Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of a and b only.
• 17N.1.hl.TZ0.9e: Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of...
• 18M.1.hl.TZ2.9a.i: Explain why ABCD is a parallelogram.
• 18M.1.hl.TZ2.9a.ii: Using vector algebra, show that...
• 18M.1.hl.TZ2.9b: Show that p = 1, q = 1 and r = 4.
• 18M.1.hl.TZ2.9c: Find the area of the parallelogram ABCD.

Algebraic and geometric approaches to multiplication by a scalar, \(kv\) .

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 14M.1.hl.TZ1.12b: Find the coordinates of M, the mid-point of [OB].
• 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}}  = \) \(\boldsymbol{a}\) and...
• 14M.1.hl.TZ1.12a: Show that the points \({\text{O}}(0,{\text{ }}0,{\text{ }}0)\),...
• 15N.1.hl.TZ0.13a: Find \(\overrightarrow {{\text{BR}}} \) in terms of \({{a}}\), \({{b}}\) and \({{c}}\).
• 15N.2.hl.TZ0.10a: find the vectors \({{a}}\) and \({{b}}\).
• 16M.1.hl.TZ1.8: O, A, B and C are distinct points such that \(\overrightarrow {{\text{OA}}}  = \) a,...
• 16M.2.hl.TZ2.9a: Show that (i)     \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)a\({|^2} +...
• 16M.2.hl.TZ2.9b: Given that...
• 17N.1.hl.TZ0.9a.i: Find, in terms of a and b \(\overrightarrow {{\text{OF}}} \).
• 17N.1.hl.TZ0.9a.ii: Find, in terms of a and b \(\overrightarrow {{\text{AF}}} \).
• 17N.1.hl.TZ0.9b.i: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\lambda \);
• 17N.1.hl.TZ0.9b.ii: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\mu \).
• 17N.1.hl.TZ0.9c: Show that \(\mu  = \frac{1}{{13}}\), and find the value of \(\lambda \).
• 17N.1.hl.TZ0.9d: Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of a and b only.
• 17N.1.hl.TZ0.9e: Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of...
• 18M.1.hl.TZ2.9a.i: Explain why ABCD is a parallelogram.
• 18M.1.hl.TZ2.9a.ii: Using vector algebra, show that...
• 18M.1.hl.TZ2.9b: Show that p = 1, q = 1 and r = 4.
• 18M.1.hl.TZ2.9c: Find the area of the parallelogram ABCD.

Algebraic and geometric approaches to magnitude of a vector, \(\left| v \right|\) .

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 10M.1.hl.TZ2.3: The three vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\) and \(\boldsymbol{c}\) are given...
• 10N.1.hl.TZ0.10: Let \(\alpha \) be the angle between the unit vectors a and b, where...
• 14M.1.hl.TZ1.12b: Find the coordinates of M, the mid-point of [OB].
• 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}}  = \) \(\boldsymbol{a}\) and...
• 14M.1.hl.TZ1.12a: Show that the points \({\text{O}}(0,{\text{ }}0,{\text{ }}0)\),...
• 14N.1.hl.TZ0.3: A point \(P\), relative to an origin \(O\), has position vector...
• 16M.1.hl.TZ1.8: O, A, B and C are distinct points such that \(\overrightarrow {{\text{OA}}}  = \) a,...
• 16M.2.hl.TZ2.9a: Show that (i)     \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)a\({|^2} +...
• 16M.2.hl.TZ2.9b: Given that...
• 17N.1.hl.TZ0.9a.i: Find, in terms of a and b \(\overrightarrow {{\text{OF}}} \).
• 17N.1.hl.TZ0.9a.ii: Find, in terms of a and b \(\overrightarrow {{\text{AF}}} \).
• 17N.1.hl.TZ0.9b.i: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\lambda \);
• 17N.1.hl.TZ0.9b.ii: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\mu \).
• 17N.1.hl.TZ0.9c: Show that \(\mu  = \frac{1}{{13}}\), and find the value of \(\lambda \).
• 17N.1.hl.TZ0.9d: Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of a and b only.
• 17N.1.hl.TZ0.9e: Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of...
• 18M.1.hl.TZ2.9a.i: Explain why ABCD is a parallelogram.
• 18M.1.hl.TZ2.9a.ii: Using vector algebra, show that...
• 18M.1.hl.TZ2.9b: Show that p = 1, q = 1 and r = 4.
• 18M.1.hl.TZ2.9c: Find the area of the parallelogram ABCD.

Algebraic and geometric approaches to position vectors \(\overrightarrow {OA} = a\) .

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 14M.1.hl.TZ1.12b: Find the coordinates of M, the mid-point of [OB].
• 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}}  = \) \(\boldsymbol{a}\) and...
• 14M.1.hl.TZ1.12a: Show that the points \({\text{O}}(0,{\text{ }}0,{\text{ }}0)\),...
• 14N.1.hl.TZ0.12a: (i)     Express \(\overrightarrow {{\text{AM}}} \) in terms of \(a\) and \(c\). (ii)     Hence...
• 14N.1.hl.TZ0.12b: (i)     Express \(\overrightarrow {{\text{RA}}} \) in terms of \(a\) and \(b\). (ii)     Show...
• 14N.1.hl.TZ0.12c: Prove that \(T\) lies on [\(BM\)].
• 15N.1.hl.TZ0.13a: Find \(\overrightarrow {{\text{BR}}} \) in terms of \({{a}}\), \({{b}}\) and \({{c}}\).
• 16M.1.hl.TZ1.8: O, A, B and C are distinct points such that \(\overrightarrow {{\text{OA}}}  = \) a,...
• 16M.2.hl.TZ2.9a: Show that (i)     \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)a\({|^2} +...
• 16M.2.hl.TZ2.9b: Given that...
• 17N.1.hl.TZ0.9a.i: Find, in terms of a and b \(\overrightarrow {{\text{OF}}} \).
• 17N.1.hl.TZ0.9a.ii: Find, in terms of a and b \(\overrightarrow {{\text{AF}}} \).
• 17N.1.hl.TZ0.9b.i: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\lambda \);
• 17N.1.hl.TZ0.9b.ii: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\mu \).
• 17N.1.hl.TZ0.9c: Show that \(\mu  = \frac{1}{{13}}\), and find the value of \(\lambda \).
• 17N.1.hl.TZ0.9d: Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of a and b only.
• 17N.1.hl.TZ0.9e: Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of...
• 18M.1.hl.TZ2.9a.i: Explain why ABCD is a parallelogram.
• 18M.1.hl.TZ2.9a.ii: Using vector algebra, show that...
• 18M.1.hl.TZ2.9b: Show that p = 1, q = 1 and r = 4.
• 18M.1.hl.TZ2.9c: Find the area of the parallelogram ABCD.

\(\overrightarrow {AB} = b - a\) .

• 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
• 14N.1.hl.TZ0.12a: (i)     Express \(\overrightarrow {{\text{AM}}} \) in terms of \(a\) and \(c\). (ii)     Hence...
• 14N.1.hl.TZ0.12b: (i)     Express \(\overrightarrow {{\text{RA}}} \) in terms of \(a\) and \(b\). (ii)     Show...
• 14N.1.hl.TZ0.12c: Prove that \(T\) lies on [\(BM\)].
• 15M.2.hl.TZ2.11c: Find the distance between the two points on the curve where each tangent is parallel to the line...
• 15N.1.hl.TZ0.13a: Find \(\overrightarrow {{\text{BR}}} \) in terms of \({{a}}\), \({{b}}\) and \({{c}}\).
• 17N.1.hl.TZ0.9a.ii: Find, in terms of a and b \(\overrightarrow {{\text{AF}}} \).