SSCE HSC Mathematics Extension 1 Scalar product of vectors: Given the two non-zero vectors a

Given the two non-zero vectors a and b, let c be the projection of a onto b - HSC - SSCE Mathematics Extension 1 - Question 6 - 2023 - Paper 1

Question 6

Given the two non-zero vectors a and b, let c be the projection of a onto b. What is the projection of 10a onto 2b? A. 2c B. 5c C. 10c D. 20c

Worked Solution & Example Answer:Given the two non-zero vectors a and b, let c be the projection of a onto b - HSC - SSCE Mathematics Extension 1 - Question 6 - 2023 - Paper 1

Step 1

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Answer

To find the projection of a vector onto another vector, we use the formula:

$ext{proj}_{ ext{b}}( ext{a}) = rac{ ext{a} ullet ext{b}}{ ext{b} ullet ext{b}} ext{b}$

Here, we need to calculate the projection of 10a onto 2b.

Calculate the projection: Using the projection formula:
- First, find the dot product:
  $ext{proj}_{2b}(10a) = rac{10a ullet (2b)}{(2b) ullet (2b)} (2b)$
- Simplifying, we have: $ext{proj}_{2b}(10a) = rac{20(a ullet b)}{4(b ullet b)} (2b)$
- This further simplifies to: $ext{proj}_{2b}(10a) = rac{20}{4} rac{(a ullet b)}{(b ullet b)} (2b) = 5 ext{proj}_{b}(a) (2b) = 10c$

Hence, the projection of 10a onto 2b is equal to 10c, which corresponds to option C.