Question 19:
--- 19(a) ---
19(a) | Finds a direction vector for the second wire.
Condone one error. | 3.4 | M1 | Direction vector for 2nd wire =
10 −10
�0 �−�100�
20 −5
and
0 0 10 20
𝒓𝒓𝟏𝟏 = �0�+𝜆𝜆�100� 𝒓𝒓𝟐𝟐 = �0 �+𝜇𝜇�−100�
0 −20 20 25
10+20𝜇𝜇
𝒓𝒓𝟐𝟐−𝒓𝒓𝟏𝟏 = �−100𝜇𝜇−100𝜆𝜆�
20+25𝜇𝜇+20𝜆𝜆
and
10+20𝜇𝜇 0 10+20𝜇𝜇 20
�−100𝜇𝜇−100𝜆𝜆�.�100�= 0 �−100𝜇𝜇−100𝜆𝜆�.�−100�= 0
20+25𝜇𝜇+20𝜆𝜆 −20 20+25𝜇𝜇+20𝜆𝜆 and2 5
−10000𝜇𝜇−10000𝜆𝜆−400 −500𝜇𝜇−400𝜆𝜆 = 0
200+400𝜇𝜇+10000𝜇𝜇+1 0 0a0n0d𝜆𝜆 +500+625𝜇𝜇+500𝜆𝜆 = 0
11025𝜇𝜇+10500𝜆𝜆+700 = 0 11025𝜇𝜇+10920𝜆𝜆+420 = 0
and
2 44
𝜆𝜆 = 3 𝜇𝜇 = −63
2 2 2
44 44 2 44 2
��10+20�− �� +�−100�− �−100� �� +�20+25�− �+20� ��
63 63 3 63 3
= 16.7 metres
Writes, in terms of a parameter, the position vector (or
coordinates) of one point on each of the two lines.
Condone use of same parameter. | 3.1a | M1
Obtains, in terms of two parameters, a correct vector
between the two lines. | 1.1b | A1
Sets up two scalar products for their and their
valid direction vectors.
𝒓𝒓𝟐𝟐−𝒓𝒓𝟏𝟏 | 1.1a | M1
Obtains correct parameter values. | 1.1b | A1
Uses full method for required distance | 1.1b | M1
Obtains correct distance to 2, 3 or 4 significant figures
with correct units.
Accept 1 significant figure if full method shown. | 3.2a | A1
Q | Marking instructions | AO | Mark | Typical solution
ALT
19(a) | Finds a direction vector for the second wire. | 3.4 | M1 | Direction vector for 2nd wire =
10 −10
�0 �−�100�
Let be a vector perpendicula2r0 to both− w5ires.
𝑥𝑥
�𝑦𝑦�
𝑧𝑧 and
0 𝑥𝑥 20 𝑥𝑥
∴ �100�.�𝑦𝑦� = 0 �−100�.�𝑦𝑦� = 0
and
−20 𝑧𝑧 25 𝑧𝑧
⇒100𝑦𝑦−20𝑧𝑧 = 0 20𝑥𝑥−100𝑦𝑦+25𝑧𝑧 = 0
and
⇒𝑧𝑧 = 5𝑦𝑦 𝑥𝑥 = −1.25𝑦𝑦
perpendicular vector is
−1.25𝑦𝑦
∴ � 𝑦𝑦 �
5𝑦𝑦
unit perpendicular vector is
−1.25
2 2 2
⇒ � 1 �÷�(−1.25) +1 +5
5
a vector from 1st line to 2nd line is
10 0 10
�0 � −�0�= �0 �
20 0 20
distance between lines is
10 −1.25 21
∴ �0 �.� 1 �÷ 4
20 5
= 16.7 metres
Forms two equations for a perpendicular vector | 3.1a | M1
Obtains two correct equations for a perpendicular vector | 1.1b | A1
Obtains a correct normal vector | 1.1b | A1
Finds the unit normal vector | 1.1a | M1
Uses full method for required distance | 1.1b | M1
Obtains correct distance to 2, 3 or 4 significant figures
with correct units.
Accept 1 significant figure if full method shown. | 3.2a | A1
--- 19(b) ---
19(b) | Suggests an improvement to the model.
Do not condone criticisms without refinements. | 3.5c | B1 | Model the wires as curves
Total | 8
TOTAL | 80