Answers
A3.7 Vectors
Data structures · A level · AQA 7517 4.2.1.4 · about 25 min
Data structures · A level · AQA 7517 4.2.1.4 · about 25 min
Vectors as lists, functions and arrows; addition, scaling, dot product and convex combination, on the robot's position.
[1 mark]What is the dot product of [2, -1, 3] and [4, 5, 1]?
[1 mark]What does this program print?
u = [2, 5] v = [6, 1] w = [0.5 * u[i] + 0.5 * v[i] for i in range(2)] print(w)
[4.0, 3.0]
With both weights 0.5 the convex combination is the midpoint of u and v.
[1 mark]For αu + βv to be a convex combination of u and v, which conditions must hold?
[1 mark]The dot product of two non-zero vectors is 0. What does that tell you?
[1 mark]The vector [2.0, 3.5, -1.0] is a member of which set?
[1 mark]What is the angle, in degrees, between the vectors [1, 0] and [1, 1]?
Two beacons are at u = [40, 20] and v = [-20, 60], in cm from the robot's start (x to the right, y forward). Write these three functions yourself, for vectors of any length, without numpy:
- add(a, b): returns a new list, the vector sum of a and b.
- scale(k, a): returns a new list, the number k times the vector a.
- dot(a, b): returns the dot product of a and b, a single number.
Then print, in this order:
1. u.v = <dot product>, which is a whole number.
2. angle = <angle> degrees, the angle between u and v rounded to a whole number with round(). You will need math.acos, math.degrees and math.sqrt.
3. w = <w>, where w = 0.25u + 0.75v, built with scale and add, printed as a Python list, for example w = [1.0, 2.0].
Finally drive to w: slide sideways by its x component (left if it is negative) and then forward by its y component.
# the two lines every program starts with: the commands, then the robot from bugbot import * connect() import math u = [40, 20] v = [-20, 60]
The hint students can ask for: The dot product multiplies matching components and adds the results. The angle comes from rearranging u.v = |u||v| cos(angle), and each length is the square root of a vector's dot product with itself. The convex combination is two scalings and an addition; its x component tells you how far to slide and which way.
# the two lines every program starts with: the commands, then the robot
from bugbot import *
connect()
import math
u = [40, 20]
v = [-20, 60]
def add(a, b):
return [a[i] + b[i] for i in range(len(a))]
def scale(k, a):
return [k * x for x in a]
def dot(a, b):
total = 0
for i in range(len(a)):
total = total + a[i] * b[i]
return total
def length(a):
return math.sqrt(dot(a, a))
print("u.v =", dot(u, v))
angle = math.degrees(math.acos(dot(u, v) / (length(u) * length(v))))
print("angle =", round(angle), "degrees")
w = add(scale(0.25, u), scale(0.75, v))
print("w =", w)
if w[0] > 0:
right(50, distance=w[0])
else:
left(50, distance=-w[0])
forward(50, distance=w[1])
Any program that meets the task's checks is marked correct in the simulator; this is one way, not the only way.