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- A line through the origin is all multiples of a vector. A plane through the origin is all multiples of two vectors added together. Any other line is one vector plus all mutiples of a second. Any other plane is one vector plus all multiples of two other vectors.
math.stackexchange.com/questions/4265977/why-lines-and-planes-through-the-origin-and-only-lines-and-planes-through-thelinear algebra - why "Lines and planes through the origin ...
Sep 2, 2021 · Geometrically, \(\mathbf{x}\) and \(\mathbf{y}\) are linearly independent if they do not lie on the same line through the origin. Notice that for any vector \(\mathbf{x}\), \(\mathbf{0}\) and \(\mathbf{x}\) are not linearly independent, that is, they are linearly dependent , since \(\mathbf{0}=0 \mathbf{x}\).
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Exercise \(\PageIndex{1}\) For each of the following pairs...
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Oct 2, 2021 · A line through the origin is all multiples of a vector. A plane through the origin is all multiples of two vectors added together. Any other line is one vector plus all mutiples of a second.
Dec 21, 2020 · Unlike a plane, a line in three dimensions does have an obvious direction, namely, the direction of any vector parallel to it. In fact a line can be defined and uniquely identified by providing one point on the line and a vector parallel to the line (in one of two possible directions).
Lines and Planes in R3. A line in R3 is determined by a point (a; b; c) on the line and a direction ~v that is parallel(1) to the line. This represents that we start at the point (a; b; c) and add all scalar multiples of the vector ~v.
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Unlike a plane, a line in three dimensions does have an obvious direction, namely, the direction of any vector parallel to it. In fact a line can be defined and uniquely identified by providing one point on the line and a vector parallel to the line (in one of two possible directions).
A line can be defined as a collection of points and as the collection of vectors that connect the origin to points on the line. Similarly, we can identify the plane \(\mathcal{P}\) as the collection of all vectors \(\mathbf{v}\) that connect the origin to points on \(\mathcal{P}\) .
Oct 27, 2024 · Suppose that n is a normal vector to a plane and \((a,b,c)\) is a point on the plane. Let \((x,y,z)\) be a general point on the plane, then \[ \langle x - a, y - b, z - c\rangle \nonumber \] is parallel to the plane, hence \[\vec{n} \cdot \langle x - a, y - b, z - c\rangle = 0.\nonumber \] This defines the equation of the plane.
