Question

1. which of the following vectors are parallel to \langle 2,4 \

angle and which point in the same direction?\

1. \langle 1,1 \

angle\

1. \langle 1,2 \

angle\

1. \langle -1,2 \

angle\

1. \langle -1,-2 \

angle\

1. \langle 10,20 \

angle\

1. \langle 3,4 \

angle\

1. \langle -5,-10 \

angle

Explanation:

Step1: Define parallel vector rule

A vector $\langle x,y
angle$ is parallel to $\langle 2,4
angle$ if $\langle x,y
angle = k\langle 2,4
angle$ for some scalar $k
eq 0$. Same direction requires $k>0$, opposite direction requires $k<0$.

Step2: Test vector $\langle1,1

angle$
Check if $\langle1,1
angle = k\langle2,4
angle$:
$2k=1 \implies k=\frac{1}{2}$; $4k=2
eq 1$. Not parallel.

Step3: Test vector $\langle1,2

angle$
Check if $\langle1,2
angle = k\langle2,4
angle$:
$2k=1 \implies k=\frac{1}{2}$; $4k=2$. Valid, $k>0$. Parallel, same direction.

Step4: Test vector $\langle-1,2

angle$
Check if $\langle-1,2
angle = k\langle2,4
angle$:
$2k=-1 \implies k=-\frac{1}{2}$; $4k=-2
eq 2$. Not parallel.

Step5: Test vector $\langle-1,-2

angle$
Check if $\langle-1,-2
angle = k\langle2,4
angle$:
$2k=-1 \implies k=-\frac{1}{2}$; $4k=-2$. Valid, $k<0$. Parallel, opposite direction.

Step6: Test vector $\langle10,20

angle$
Check if $\langle10,20
angle = k\langle2,4
angle$:
$2k=10 \implies k=5$; $4k=20$. Valid, $k>0$. Parallel, same direction.

Step7: Test vector $\langle3,4

angle$
Check if $\langle3,4
angle = k\langle2,4
angle$:
$2k=3 \implies k=\frac{3}{2}$; $4k=6
eq 4$. Not parallel.

Step8: Test vector $\langle-5,-10

angle$
Check if $\langle-5,-10
angle = k\langle2,4
angle$:
$2k=-5 \implies k=-\frac{5}{2}$; $4k=-10$. Valid, $k<0$. Parallel, opposite direction.

Answer:

Parallel to $\langle 2,4
angle$:

• Same direction: 2. $\langle 1,2

angle$, 5. $\langle 10,20
angle$

• Opposite direction: 4. $\langle -1,-2

angle$, 7. $\langle -5,-10
angle$
Not parallel: 1. $\langle 1,1
angle$, 3. $\langle -1,2
angle$, 6. $\langle 3,4
angle$