DP Mathematics HL Questionbank

• Syllabus

Topic 4 - Core: Vectors

Sub sections and their related questions

4.1

• 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 .
• 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...
• 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 $\overrightarrow {{\text{AC}}}$.
• 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...
• 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 ${{c}}$.
• 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.2.hl.TZ1.11a: Show that the two submarines would collide at a point P and write down the coordinates of P.
• 18M.2.hl.TZ1.11b.i: Show that submarine B travels in the same direction as originally planned.
• 18M.2.hl.TZ1.11b.ii: Find the value of t when submarine B passes through P.
• 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.

4.2

• 12M.2.hl.TZ1.13a: Show that AB = AC and that ${\rm{B\hat AC}} = 60^\circ$.
• 12M.2.hl.TZ1.13f: A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around...
• 12M.1.hl.TZ2.2: Find the values of x for which the vectors...
• 12N.2.hl.TZ0.13e: Let Y be a point on ${L_1}$ and Z be a point on ${L_2}$ such that XY is perpendicular to YZ...
• 08M.1.hl.TZ2.10: Given any two non-zero vectors a and b , show that $|$a $\times$ b${|^2}$ =...
• 08N.1.hl.TZ0.10: Three distinct non-zero vectors are given by $\overrightarrow {{\text{OA}}}$ = a,...
• 08N.2.hl.TZ0.4: The angle between the vector a = i − 2j + 3k and the vector b = 3i − 2j + mk is 30° . Find the...
• 11M.1.hl.TZ2.6b: Hence prove that ${\rm{A\hat CB}}$ is a right angle.
• 13M.2.hl.TZ1.11c: Find the acute angle between ${L_1}$ and ${L_2}$.
• 13M.2.hl.TZ1.11e: Let S be a point on ${L_2}$ such that...
• 10M.1.hl.TZ1.6: Consider the vectors $\overrightarrow {{\text{OA}}}$ = a, $\overrightarrow {{\text{OB}}}$ =...
• 10M.1.hl.TZ2.12a: Consider the vectors a = 6i + 3j + 2k, b = −3j + 4k. (i)     Find the cosine of the angle...
• 10M.1.hl.TZ2.12b: Given two vectors p and q, (i)     show that p$\cdot$p = $|$p${|^2}$; (ii)     hence,...
• 10N.2.hl.TZ0.9: Consider the vectors a $= \sin (2\alpha )$i $- \cos (2\alpha )$j + k and b...
• 13M.1.hl.TZ2.11a: (i)     Find the lengths of the sides of the triangle. (ii)     Find $\cos {\rm{B\hat AC}}$.
• 11N.1.hl.TZ0.12a: For non-zero vectors ${\boldsymbol{a}}$ and ${\boldsymbol{b}}$, show that (i)     if...
• 11N.1.hl.TZ0.12b: The points A, B and C have position vectors ${\boldsymbol{a}}$, ${\boldsymbol{b}}$ and...
• 11M.1.hl.TZ1.4b: Hence prove that angle ${\text{A}}\hat {\rm{B}}{\text{C}}$ is a right angle.
• 09M.2.hl.TZ2.2: Given that...
• 14M.1.hl.TZ1.12g: (i)     Find the angle ${\rm{O\hat DA}}$. (ii)     State what this tells you about the solid...
• 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that $\overrightarrow {{\text{PQ}}}  =$ $\boldsymbol{a}$ and...
• 13N.2.hl.TZ0.7a: show that $3{\sin ^2}\theta  - 7\sin \theta  + 2 = 0$;
• 15N.2.hl.TZ0.10b: Roderick is at a point ${\text{C}}(11,{\text{ }}9)$. During Ed’s walk from $P$ to $Q$...
• 16M.1.hl.TZ1.11a: Find the cosine of the angle between the two planes in the form $\sqrt {\frac{p}{q}}$ where...
• 17M.1.hl.TZ1.5b: By using a suitable scalar product of two vectors, determine whether ${\rm{A\hat BC}}$ is acute...
• 17M.2.hl.TZ2.9g: Find the acute angle between the planes Π$_2$ and Π$_3$.
• 18M.1.hl.TZ2.1: The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ. Find cos θ.

4.3

• 12M.2.hl.TZ1.4: The planes $2x + 3y - z = 5$ and $x - y + 2z = k$ intersect in the line...
• 12M.2.hl.TZ1.13d: Find the vector equation of L , the line of intersection of ${\Pi _1}$ and ${\Pi _2}$ , and...
• 12M.2.hl.TZ1.13e: A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around...
• 12M.2.hl.TZ2.11b: Given that the system of equations can be solved, find the solutions in the form of a vector...
• 12N.2.hl.TZ0.13b: The planes ${\pi _1}$ and ${\pi _2}$ intersect in the line ${L_1}$ . Show that the vector...
• 11M.2.hl.TZ2.11e: Write down the vector equation of the line through the origin (0, 0, 0) that is perpendicular to...
• 11M.2.hl.TZ2.11f: Hence find the point on the plane that is closest to the origin.
• SPNone.2.hl.TZ0.10b: (i)     Find the vector equation of the line ${L_1}$ containing the points A and B. (ii)    ...
• 13M.2.hl.TZ1.11a: Find a vector equation for the line, ${L_1}$, which passes through the points P and Q. The...
• 13M.2.hl.TZ1.11c: Find the acute angle between ${L_1}$ and ${L_2}$.
• 13M.2.hl.TZ1.11d: Let S be a point on ${L_2}$ such that...
• 13M.2.hl.TZ1.11e: Let S be a point on ${L_2}$ such that...
• 13M.1.hl.TZ2.11d: Find a vector equation of (AB).
• 13M.1.hl.TZ2.11e: The point D on (AB) is such that $\overrightarrow {{\text{OD}}}$ is perpendicular to...
• 11N.2.hl.TZ0.13a: Find the vector equation of L, the line of intersection of ${\Pi _1}$ and ${\Pi _2}$.
• 11M.2.hl.TZ1.10: Port A is defined to be the origin of a set of coordinate axes and port B is located at the point...
• 09N.2.hl.TZ0.2: The vector equation of line $l$ is given as...
• 09N.2.hl.TZ0.11: (a)     Find the coordinates of the point $A$ on ${l_1}$ and the point $B$ on ${l_2}$...
• 09M.2.hl.TZ1.5: Find the angle between the lines $\frac{{x - 1}}{2} = 1 - y = 2z$ and $x = y = 3z$ .
• 09M.2.hl.TZ1.11: (a)     Find the coordinates of P when $t = 0$ . (b)     Show that P moves along the line...
• 14M.1.hl.TZ1.12d: Find a vector equation of the line $L$, through M, perpendicular to the plane ${\mathit{\Pi }}$.
• 14M.1.hl.TZ2.12a: Given the points A(1, 0, 4), B(2, 3, −1) and C(0, 1, − 2) , find the vector equation of the line...
• 13N.1.hl.TZ0.11d: A second plane ${\Pi _2}$ is defined by the Cartesian equation ${\Pi _2}:4x - y - z = 4$....
• 13N.2.hl.TZ0.9: A line ${L_1}$ has equation r =...
• 14N.2.hl.TZ0.5a: Find the Cartesian equation of $\pi$.
• 15M.2.hl.TZ2.13a: Show that the lines ${L_1}$ and ${L_2}$ are skew.
• 15M.2.hl.TZ2.13b: Find the acute angle between the lines ${L_1}$ and ${L_2}$.
• 15N.1.hl.TZ0.13b: (i)     Find a vector equation of the line that passes through $B$ and $R$ in terms of...
• 15N.1.hl.TZ0.13c: Show that the line segment [$CP$] also includes the point $G$.
• 15N.2.hl.TZ0.10a: find the vectors ${{a}}$ and ${{b}}$.
• 16M.1.hl.TZ2.10a: Show that $L$ is not perpendicular to $\Pi$.
• 16M.1.hl.TZ2.10b: Given that $L$ lies in the plane $\Pi$, find the value of $p$ and the value of $q$.
• 16M.1.hl.TZ1.11b: (i)     Show that $L$ has direction...
• 16N.1.hl.TZ0.8a: find the value of $a$;
• 16N.2.hl.TZ0.2: Find the acute angle between the planes with equations $x + y + z = 3$ and $2x - z = 2$.
• 17M.2.hl.TZ2.9a: Find the vector equation of the line (BC).
• 17N.1.hl.TZ0.2a: Find a vector equation of the line L passing through the points A and B.
• 18M.1.hl.TZ2.9d: Find the vector equation of the straight line passing through M and normal to the plane \(\Pi...

4.4

• 12N.1.hl.TZ0.9a: Find the coordinates of the common point of the paths of the two boats.
• 12N.1.hl.TZ0.9b: Show that the boats do not collide.
• 12N.2.hl.TZ0.13c: The line ${L_2}$ has Cartesian equation $5 - x = y + 3 = 2 - 2z$ . The lines ${L_1}$ and...
• 08M.1.hl.TZ2.11: Consider the points A(1, −1, 4), B (2, − 2, 5) and O(0, 0, 0). (a)     Calculate the cosine of...
• SPNone.2.hl.TZ0.10b: (i)     Find the vector equation of the line ${L_1}$ containing the points A and B. (ii)    ...
• 13M.2.hl.TZ1.11b: Show that ${L_1}$ and ${L_2}$ intersect at the point R(1, 2, 4).
• 14M.1.hl.TZ2.12b: The line ${L_2}$ has Cartesian equation...
• 15M.2.hl.TZ2.13a: Show that the lines ${L_1}$ and ${L_2}$ are skew.
• 15N.1.hl.TZ0.13b: (i)     Find a vector equation of the line that passes through $B$ and $R$ in terms of...
• 15N.1.hl.TZ0.13c: Show that the line segment [$CP$] also includes the point $G$.
• 15N.2.hl.TZ0.10b: Roderick is at a point ${\text{C}}(11,{\text{ }}9)$. During Ed’s walk from $P$ to $Q$...
• 17M.2.hl.TZ2.9b: Determine whether or not the lines (OA) and (BC) intersect.
• 18M.2.hl.TZ1.11c.i: Find an expression for the distance between the two submarines in terms of t.
• 18M.2.hl.TZ1.11c.ii: Find the value of t when the two submarines are closest together.
• 18M.2.hl.TZ1.11c.iii: Find the distance between the two submarines at this time.

4.5

• 12N.2.hl.TZ0.13d: Determine a Cartesian equation of the plane ${\pi _3}$ containing both lines ${L_1}$ and...
• 08M.1.hl.TZ2.10: Given any two non-zero vectors a and b , show that $|$a $\times$ b${|^2}$ =...
• 11M.2.hl.TZ2.11b: The vector product...
• 11M.2.hl.TZ2.11c: Hence calculate the area of parallelogram PQRS.
• 09M.1.hl.TZ1.8: A triangle has vertices A(1, −1, 1), B(1, 1, 0) and C(−1, 1, −1) . Show that the area of the...
• SPNone.1.hl.TZ0.8: The vectors a , b , c satisfy the equation a + b + c = 0 . Show that a $\times$ b = b...
• SPNone.2.hl.TZ0.10a: (i)     Find $\overrightarrow {{\text{OA}}}  \times \overrightarrow {{\text{OB}}}$ . (ii)    ...
• SPNone.2.hl.TZ0.13e: Consider the points ${\text{A}}\left( {a{\text{ }},{\text{ }}f(a)} \right)$ ,...
• 13M.1.hl.TZ1.2a: Find $\overrightarrow {{\text{AB}}}  \times \overrightarrow {{\text{AC}}}$.
• 13M.1.hl.TZ1.2b: Hence find the area of the triangle ABC.
• 10M.1.hl.TZ2.12a: Consider the vectors a = 6i + 3j + 2k, b = −3j + 4k. (i)     Find the cosine of the angle...
• 10N.2.hl.TZ0.9: Consider the vectors a $= \sin (2\alpha )$i $- \cos (2\alpha )$j + k and b...
• 10N.2.hl.TZ0.12: The diagram shows a cube OABCDEFG.     Let O be the origin, (OA) the x-axis, (OC) the y-axis...
• 13M.1.hl.TZ2.11b: (i)     Show that $\overrightarrow {{\text{BC}}}  \times \overrightarrow {{\text{CA}}}  =$ −7i...
• 11N.1.hl.TZ0.12a: For non-zero vectors ${\boldsymbol{a}}$ and ${\boldsymbol{b}}$, show that (i)     if...
• 11N.1.hl.TZ0.12b: The points A, B and C have position vectors ${\boldsymbol{a}}$, ${\boldsymbol{b}}$ and...
• 11M.1.hl.TZ1.11b: Find the Cartesian equation of the plane $\prod$ that contains the face ABC.
• 13N.1.hl.TZ0.11b: Find an exact value for the area of the triangle ABC.
• 13N.1.hl.TZ0.11a: Find the vector $\overrightarrow {{\text{CA}}}  \times \overrightarrow {{\text{CB}}}$.
• 15N.1.hl.TZ0.13d: The coordinates of the points $A$, $B$ and $C$ are $(1,{\text{ }}3,{\text{ }}1)$,...
• 16M.2.hl.TZ1.1a: Find $\overrightarrow {{\text{OA}}}  \times \overrightarrow {{\text{OB}}}$.
• 16M.2.hl.TZ1.1b: Hence find the area of the triangle OAB.
• 16N.1.hl.TZ0.4a: Find a $\times$ b.
• 17M.1.hl.TZ1.5a: Find the area of the parallelogram ABCD.
• 17M.2.hl.TZ2.7: Given that a $\times$ b $=$ b $\times$ c $\ne$ 0 prove that a $+$ c $=$...

4.6

• 12M.2.hl.TZ1.4: The planes $2x + 3y - z = 5$ and $x - y + 2z = k$ intersect in the line...
• 12M.2.hl.TZ1.13b: Find the Cartesian equation of $\Pi$, the plane passing through A, B, and C.
• 12M.2.hl.TZ1.13c(i)(ii):  (i)     Find the Cartesian equation of ${\Pi _1}$ , the plane perpendicular to (AB)...
• 12M.2.hl.TZ2.11c: The plane $\div$ is parallel to both the line in part (b) and the line...
• 12N.2.hl.TZ0.13d: Determine a Cartesian equation of the plane ${\pi _3}$ containing both lines ${L_1}$ and...
• 08M.1.hl.TZ2.11: Consider the points A(1, −1, 4), B (2, − 2, 5) and O(0, 0, 0). (a)     Calculate the cosine of...
• 11M.2.hl.TZ2.11d: Find the Cartesian equation of the plane, ${\prod _1}$ , containing the parallelogram PQRS.
• SPNone.2.hl.TZ0.10a: (i)     Find $\overrightarrow {{\text{OA}}}  \times \overrightarrow {{\text{OB}}}$ . (ii)    ...
• 10M.1.hl.TZ1.3: (a)     Show that the two planes $${\pi _1}:x + 2y - z = 1$$ $${\pi _2}:x + z =  - 2$$ are...
• 10M.2.hl.TZ1.11: A plane $\pi$ has vector equation r = (−2i + 3j − 2k) + $\lambda$(2i + 3j + 2k) + \(\mu...
• 10M.1.hl.TZ2.12a: Consider the vectors a = 6i + 3j + 2k, b = −3j + 4k. (i)     Find the cosine of the angle...
• 13M.1.hl.TZ2.11c: Find the Cartesian equation of the plane containing the triangle ABC.
• 11N.2.hl.TZ0.13b: Show that the plane ${\Pi _3}$ which is perpendicular to ${\Pi _1}$ and contains L, has...
• 11M.1.hl.TZ1.11b: Find the Cartesian equation of the plane $\prod$ that contains the face ABC.
• 09N.2.hl.TZ0.2: The vector equation of line $l$ is given as...
• 09N.2.hl.TZ0.11: (a)     Find the coordinates of the point $A$ on ${l_1}$ and the point $B$ on ${l_2}$...
• 14M.1.hl.TZ1.12c: Show that an equation of the plane ${\mathit{\Pi }}$, containing the square OABC, is...
• 14M.1.hl.TZ2.12d: (i)     Find the value of $k$. (ii)     Find the point of intersection P of the line ${L_3}$...
• 13N.1.hl.TZ0.11c: Show that the Cartesian equation of the plane ${\Pi _1}$, containing the triangle ABC, is...
• 14M.1.hl.TZ2.12c: Find the Cartesian equation of the plane ${\Pi _1}$.
• 14N.2.hl.TZ0.5a: Find the Cartesian equation of $\pi$.
• 16N.1.hl.TZ0.4b: Hence find the Cartesian equation of the plane containing the vectors a and b, and passing...
• 17M.2.hl.TZ1.7: Find the Cartesian equation of plane Π containing the points...
• 17M.2.hl.TZ2.9c: Find the Cartesian equation of the plane Π$_1$, which passes through C and is perpendicular to...
• 17M.2.hl.TZ2.9d: Show that the line (BC) lies in the plane Π$_1$.
• 17M.2.hl.TZ2.9e: Verify that 2j + k is perpendicular to the plane Π$_2$.
• 17M.2.hl.TZ2.9f: Find a vector perpendicular to the plane Π$_3$.
• 18M.1.hl.TZ1.10a: Find the Cartesian equation of the plane ${\Pi _1}$, passing through the points A , B and D.
• 18M.1.hl.TZ1.10c: Find the Cartesian equation of ${\Pi _3}$.
• 18M.1.hl.TZ2.9e: Find the Cartesian equation of $\Pi$.

4.7

• 12M.2.hl.TZ1.4: The planes $2x + 3y - z = 5$ and $x - y + 2z = k$ intersect in the line...
• 12M.2.hl.TZ2.11d: The z-axis meets the plane $\div$ at the point P. Find the coordinates of P.
• 12M.2.hl.TZ2.11e: Find the angle between the line $\frac{{x - 2}}{3} = \frac{{y + 5}}{4} = \frac{z}{2}$ and the...
• 12N.2.hl.TZ0.13a: Find the angle between the planes ${\pi _1}$and ${\pi _2}$ .
• 08M.1.hl.TZ1.11: The points A, B, C have position vectors i + j + 2k , i + 2j + 3k , 3i + k respectively and lie...
• 08M.2.hl.TZ1.3: A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector...
• 08M.2.hl.TZ1.5: Find the vector equation of the line of intersection of the three planes represented by the...
• 08N.2.hl.TZ0.10: (a)     Write the vector equations of the following lines in parametric...
• 11M.2.hl.TZ2.11g: A second plane, ${\prod _2}$ , has equation x − 2y + z = 3. Calculate the angle between the two...
• 10M.2.hl.TZ1.11: A plane $\pi$ has vector equation r = (−2i + 3j − 2k) + $\lambda$(2i + 3j + 2k) + \(\mu...
• 10N.1.hl.TZ0.7: Consider the plane with equation $4x - 2y - z = 1$ and the line given by the parametric...
• 10N.2.hl.TZ0.12: The diagram shows a cube OABCDEFG.     Let O be the origin, (OA) the x-axis, (OC) the y-axis...
• 11N.2.hl.TZ0.13c: The point P has coordinates (−2, 4, 1) , the point Q lies on ${\Pi _3}$ and PQ is perpendicular...
• 09M.2.hl.TZ1.7: (a)     If $a = 4$ find the coordinates of the point of intersection of the three...
• 14M.1.hl.TZ1.12f: Find the coordinates of E, the reflection of the point D in the plane ${\mathit{\Pi }}$.
• 14M.1.hl.TZ2.12d: (i)     Find the value of $k$. (ii)     Find the point of intersection P of the line ${L_3}$...
• 13N.1.hl.TZ0.11f: Find conditions on $\alpha$ and $\beta$ if the plane ${\Pi _3}$ does not intersect with...
• 14M.1.hl.TZ1.12e: Find the coordinates of D, the point of intersection of the line $L$ with the plane whose...
• 14M.1.hl.TZ2.12c: Find the Cartesian equation of the plane ${\Pi _1}$.
• 13N.1.hl.TZ0.11e: Find the value of $\alpha$ if all three planes contain ${L_1}$.
• 14N.2.hl.TZ0.1: Consider the two planes      ${\pi _1}:4x + 2y - z = 8$      \({\pi _2}:x + 3y + 3z =...
• 14N.2.hl.TZ0.5b: The line ${l_3}$ passing through the point $(4,{\text{ }}0,{\text{ }}8)$ is perpendicular to...
• 16M.1.hl.TZ2.1: The following system of equations represents three planes in space. \[x + 3y + z =  -...
• 16M.1.hl.TZ2.10c: (i)     Show that $p =  - 2$. (ii)     If $L$ intersects $\Pi$ at $z =  - 1$, find the...
• 16M.1.hl.TZ1.11c: Given the vector $\overrightarrow {{\text{AB}}}$ is perpendicular to $L$ find the value of...
• 16M.1.hl.TZ1.11d: Show that ${\text{AB}} = 3\sqrt 2$.
• 16M.1.hl.TZ1.11e: Find the coordinates of the two possible positions of $P$.
• 16N.1.hl.TZ0.1: Find the coordinates of the point of intersection of the planes defined by the equations...
• 16N.1.hl.TZ0.8b: determine the coordinates of the point of intersection P.
• 17N.1.hl.TZ0.2b: Find the coordinates of the point of intersection of the line L with the plane Π.
• 18M.1.hl.TZ1.10b: Find the angle between the faces ABD and BCD.
• 18M.1.hl.TZ1.10d: Show that P is the midpoint of AD.
• 18M.1.hl.TZ1.10e: Find the area of the triangle OPQ.
• 18M.1.hl.TZ2.9f.i: Find the coordinates of X, Y and Z.
• 18M.1.hl.TZ2.9f.ii: Find YZ.