List any vectors that are parallel to each other and any...

List any vectors that are parallel to each other and any vectors that are perpendicular to each other: $\vec{v}_{1}=\vec{i}-2 \vec{j} \quad \vec{v}_{2}=2 \vec{i}+4 \vec{j}$ $\vec{v}_{3}=3 \vec{i}+1.5 \vec{j} \quad \vec{v}_{4}=-1.2 \vec{i}+2.4 \vec{j}$ $\vec{v}_{5}=-5 \vec{i}-2.5 \vec{j} \quad \vec{v}_{6}=12 \vec{i}-12 \vec{j}$ $\vec{v}_{7}=4 \vec{i}+2 \vec{j} \quad \vec{v}_{8}=3 \vec{i}-6 \vec{j}$ $\vec{v}_{9}=0.70 \vec{i}-0.35 \vec{j}$

List any vectors that are parallel to each other and any vectors that are perpendicular to each other:
$\vec{v}_{1}=\vec{i}-2 \vec{j} \quad \vec{v}_{2}=2 \vec{i}+4 \vec{j}$
$\vec{v}_{3}=3 \vec{i}+1.5 \vec{j} \quad \vec{v}_{4}=-1.2 \vec{i}+2.4 \vec{j}$
$\vec{v}_{5}=-5 \vec{i}-2.5 \vec{j} \quad \vec{v}_{6}=12 \vec{i}-12 \vec{j}$
$\vec{v}_{7}=4 \vec{i}+2 \vec{j} \quad \vec{v}_{8}=3 \vec{i}-6 \vec{j}$
$\vec{v}_{9}=0.70 \vec{i}-0.35 \vec{j}$

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Key Concepts

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is essential for determining the angle between two vectors; if the dot product is zero, the vectors are perpendicular, and if one vector is a scalar multiple of the other, they are parallel.

Perpendicular Vectors

Perpendicular (or orthogonal) vectors are vectors that intersect at a right angle (90 degrees). In mathematical terms, two nonzero vectors are perpendicular if their dot product equals zero, meaning they have no component in common along the same direction.

Parallel Vectors

Parallel vectors are vectors that have the same or exactly opposite direction. This occurs when one vector is a scalar multiple of the other, meaning each of its components is multiplied by the same scalar, resulting in vectors that lie along the same line in space.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation changes the magnitude of the vector and can also reverse its direction if the scalar is negative. Scalar multiplication is the fundamental operation used to establish the parallelism of vectors.

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