• 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 $\overrightarrow {{\text{AF}}}$.