GEOMETRY-accel.
WORKSHEET – Vectors & Translations
April 23, 2014
Write all answers on this sheet. Where necessary, do work on separate graph paper and attach.
⃑⃑⃑⃑⃑ and |𝐴𝐵
⃑⃑⃑⃑⃑ |.
1. Points 𝐴 and 𝐵 are given. Find 𝐴𝐵
a. 𝐴 (1, 1); 𝐵 (5, 4)
b. 𝐴 (0, 5); 𝐵 (−3, 2)
c. 𝐴 (3, 5); 𝐵 (−1, 7)
2. Name two vectors parallel to ⟨3, −8⟩. Name a vector perpendicular to ⟨3, −8⟩.
3.
a. The vectors ⟨8, 6⟩ and ⟨12, 𝑘⟩ are parallel. Find 𝑘.
b. The vectors ⟨6, 𝑘⟩ and ⟨4, −3⟩ are perpendicular. Find 𝑘.
4. Find each vector sum; illustrate using a vector diagram.
a. ⟨2, 1⟩ + ⟨3, 6⟩
b. ⟨−8, 2⟩ − ⟨2, −3⟩
c. ⟨1, −4⟩ + 2⟨−3, 2⟩
⃑⃑⃑⃑⃑ = ⟨−1, 5⟩ and 𝐾𝑌
⃑⃑⃑⃑⃑ = ⟨7, 3⟩. What
5. Make a drawing showing an object being pulled by two forces, 𝐾𝑋
single force has the same effect as these two forces acting together? What is the magnitude of this
force? Illustrate with a vector diagram.
6. (continuation) What happens if ⃑⃑⃑⃑⃑
𝐾𝑋 = ⟨2, −3⟩ and ⃑⃑⃑⃑⃑
𝐾𝑌 = ⟨−2, 3⟩? Illustrate.
7. Suppose two non-vertical vectors ⟨𝑎, 𝑏⟩ and ⟨𝑐, 𝑑⟩ are perpendicular.
a. Use slopes to show that
𝑏𝑑
𝑎𝑐
= −1, then show that 𝑎𝑐 + 𝑏𝑑 = 0.
b. The expression 𝑎𝑐 + 𝑏𝑑 is called the dot product of vectors ⟨𝑎, 𝑏⟩ and ⟨𝑐, 𝑑⟩. Complete this
statement: If vectors ⟨𝑎, 𝑏⟩ and ⟨𝑐, 𝑑⟩ are perpendicular, then their dot product _____?____.
c. Find the dot product of ⟨3, −2⟩ and ⟨2, 5⟩.
d. Find the dot product of ⟨−6, 2⟩ and ⟨1, 3⟩.
8.
a. If 𝑇: (0, 0) → (5, 1), then 𝑇: (3, 3) → (_____, _____)
b. If 𝑇: (1, 1) → (3, 0), then 𝑇: (0, 0) → (_____, _____)
c. If 𝑇: (−2, 3) → (2, 6), then 𝑇: (_____, _____) → (0, 0)
d. The image of 𝑃 (−1, 5) under a translation is 𝑃′ (5, 7). What is the pre-image of 𝑃?
e. What is another way of writing a translation defined by the vector ⟨11, −4⟩?
9. Answer true (T) or false (F) for each of the following statements about transformations:
a. If 𝑝 is the translated image of a different line 𝑞, then 𝑝 is parallel to 𝑞.
b. It is possible for a translation to map a line 𝑝 onto a perpendicular line 𝑞.
c. It is possible for a reflection to map a line 𝑝 onto a perpendicular line 𝑞.
d. If a translation maps ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹, and a translation maps ∆𝐷𝐸𝐹 onto ∆𝐺𝐻𝐾, then
there is a translation that maps ∆𝐴𝐵𝐶 onto ∆𝐺𝐻𝐾.
e. If a translation maps ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹, then 𝐴𝐷 = 𝐵𝐸 = 𝐶𝐹.
10. Which of the three numbered triangles is NOT the
image by a reflection of the triangle at left?
1
2
3