By Nitecki Z.
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Additional resources for Calculus in 3D. Geometry, vectors, and multivariate calculus
D) Show that a collection of three position vectors in R3 is linearly independent precisely if the corresponding points determine a plane in space that does not pass through the origin. (e) Show that any collection of four or more vectors in R3 is linearly dependent. 3. 3 Lines in Space Parametrization of Lines A line in the plane is the locus of a “linear” equation in the rectangular coordinates x and y Ax + By = C where A, B and C are real constants with at least one of A and B nonzero. 14) where the slope m is the tangent of the angle the line makes with the horizontal and the y-intercept b is the ordinate (signed height) of its intersection with the y-axis.
V corresponds to P if the arrow OP from the origin to → → P is a representation of − v : that is, − v is the vector representing that → 3 displacement of R which moves the origin to P ; we refer to − v as the position vector of P . We shall make extensive use of the correspondence between vectors and points, often denoting a point by its position vector → − p ∈ R3 . Furthermore, using rectangular coordinates we can formulate a numerical specification of vectors in which addition and multiplication by scalars is −−→ → very easy to calculate: if − v = OP and P has rectangular coordinates → (x, y, z), we identify the vector − v with the triple of numbers (x, y, z) and → − write v = (x, y, z).
P C (t) = − 2 2 To find the intersection of ℓA with ℓB , we need to solve the vector equation − → → p A (r) = − p B (s) − → − −c is which, written in terms of → a , b and → − → s→ − r→ r→ s− → (1 − r)− a + b + − c = → a + (1 − s) b + − c. 2 2 2 2 Assuming the triangle △ABC is nondegenerate—which is to say, none of → − → − → a , b or − c can be expressed as a linear combination of the others—this equality can only hold if corresponding coefficients on the two sides are equal. This leads to three equations in two unknowns s (1 − r) = 2 r = (1 − s) 2 r s = .