Collinear vectors

Consider two vectors v⃗\vec v and w⃗\vec w. We say that the vectors are collinear, written

v⃗∥w⃗\vec v \parallel \vec w

if there is a scalar cc such that

v⃗=c⋅w⃗\boxed{\vec v = c \cdot \vec w}

(or if there is a scalar dd with w⃗=d⋅v⃗\vec w = d \cdot \vec v). How do the arrows of collinear vectors look like? Well, as multiplying a vector with a scalar simply stretches the arrow by cc, the arrows of collinear vectors are parallel. As cc can also be negative, the arrows can also point in opposite directions.

How do we check if two vectors are collinear? We have to compare their components and check if there is a single scalar cc such that

vx=c⋅wxvy=c⋅wyvz=c⋅wz\boxed{ \begin{array}{lcl} v_x & = & c\cdot w_x \\ v_y & = & c\cdot w_y \\ v_z & = & c\cdot w_z \\ \end{array} }
Exercise 1

Are the following vectors collinear?

  1. u⃗=(1−1.55)\vec{u} = \left(\begin{array}{r} 1\\ -1.5\\ 5 \end{array}\right) and v⃗=(2−39)\vec{v} = \left(\begin{array}{r} 2\\ -3\\ 9 \end{array}\right)

  2. u⃗=(1−1.55)\vec{u} = \left(\begin{array}{r} 1\\ -1.5\\ 5 \end{array}\right) and v⃗=(2−310)\vec{v} = \left(\begin{array}{r} 2\\ -3\\ 10 \end{array}\right)

  3. u⃗=(−423)\vec{u} = \left(\begin{array}{r} -4\\ 2\\ 3 \end{array}\right) and v⃗=(6−3−4.5)\vec{v} = \left(\begin{array}{r} 6\\ -3\\ -4.5 \end{array}\right)

  4. u⃗=(804)\vec{u} = \left(\begin{array}{r} 8\\ 0\\ 4 \end{array}\right) and v⃗=(402)\vec{v} = \left(\begin{array}{r} 4\\ 0\\ 2 \end{array}\right)

Solution
  1. no
  2. yes (u⃗=0.5v⃗\vec u = 0.5 \vec v or v⃗=2u⃗\vec v=2\vec u)
  3. yes (u⃗=−23v⃗\vec u=-\frac{2}{3}\vec v oder v⃗=−1.5u⃗\vec v=-1.5\vec u)
  4. yes (u⃗=2v⃗\vec u=2\vec v oder v⃗=0.5u⃗\vec v=0.5\vec u)