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Consider the position vectors @ = 1î −3ĵ+2k and b = −1î÷2ĵ– 2 k. (i) Find the lengths of a and b. (ii) Find the unit vectors êa and ê associated respectively with a and b. (iii) Use the cross product to find a vector c that is perpendicular to the plane passing through the origin and containing a and b. Then find the associated unit vector êc. บ (iv) A set of vectors {v₁, V2, ..., V₁} is said to form an orthogonal set if the vectors are mutually orthogonal. That is, they must satisfy = 0, ij V₁ • V j # 0, i = j. If a set of unit vectors satisfies the conditions above, the vectors are said to form an orthonormal basis. A set of vectors is said to be linearly independent if the vector equation n Σ ci v₁ = 0 has only the trivial solution c₂ = 0, Vi. i=1 Show that the set of vectors {a, b, c} are linearly independent but not orthogonal. Conse- quently, what can be said about the basis {êa, êb, êc}? (v) Find the equation of the plane passing through the terminal points of the position vectors a, b, and c. (vi) (A) A vector v can be decomposed into components that are perpendicular v to some specified direction, such that v = v ₁ + || With the aid of a diagram explain the definitions and parallel v ༅༎ ข.ข ข = ພ พ v.w v = V ข ||W||2 where w is a vector in the specified direction. (B) Find a and ዐ with W = ༅།། b. (vii) Evaluate the scalar triple product, c. (a × b) then multiply the result by 1/2. Give a geomet- ric/physical interpretation of the resultant value.

Consider the position vectors @ = 1î −3ĵ+2k and b = −1î÷2ĵ– 2 k. (i) Find the lengths of a and b. (ii) Find the unit vectors êa and ê associated respectively with a and b. (iii) Use the cross product to find a vector c that is perpendicular to the plane passing through the origin and containing a and b. Then find the associated unit vector êc. บ (iv) A set of vectors {v₁, V2, ..., V₁} is said to form an orthogonal set if the vectors are mutually orthogonal. That is, they must satisfy = 0, ij V₁ • V j # 0, i = j. If a set of unit vectors satisfies the conditions above, the vectors are said to form an orthonormal basis. A set of vectors is said to be linearly independent if the vector equation n Σ ci v₁ = 0 has only the trivial solution c₂ = 0, Vi. i=1 Show that the set of vectors {a, b, c} are linearly independent but not orthogonal. Conse- quently, what can be said about the basis {êa, êb, êc}? (v) Find the equation of the plane passing through the terminal points of the position vectors a, b, and c. (vi) (A) A vector v can be decomposed into components that are perpendicular v to some specified direction, such that v = v ₁ + || With the aid of a diagram explain the definitions and parallel v ༅༎ ข.ข ข = ພ พ v.w v = V ข ||W||2 where w is a vector in the specified direction. (B) Find a and ዐ with W = ༅།། b. (vii) Evaluate the scalar triple product, c. (a × b) then multiply the result by 1/2. Give a geomet- ric/physical interpretation of the resultant value.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.1: Inner Product Spaces
Problem 7AEXP
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Consider the position vectors @ = 1î −3ĵ+2k and b = −1î÷2ĵ– 2 k. (i) Find the lengths of a and b. (ii) Find the unit vectors êa and ê associated respectively with a and b. (iii) Use the cross product to find a vector c that is perpendicular to the plane passing through the origin and containing a and b. Then find the associated unit vector êc. บ (iv) A set of vectors {v₁, V2, ..., V₁} is said to form an orthogonal set if the vectors are mutually orthogonal. That is, they must satisfy = 0, ij V₁ • V j # 0, i = j. If a set of unit vectors satisfies the conditions above, the vectors are said to form an orthonormal basis. A set of vectors is said to be linearly independent if the vector equation n Σ ci v₁ = 0 has only the trivial solution c₂ = 0, Vi. i=1 Show that the set of vectors {a, b, c} are linearly independent but not orthogonal. Conse- quently, what can be said about the basis {êa, êb, êc}? (v) Find the equation of the plane passing through the terminal points of the position vectors a, b, and c. (vi) (A) A vector v can be decomposed into components that are perpendicular v to some specified direction, such that v = v ₁ + || With the aid of a diagram explain the definitions and parallel v ༅༎ ข.ข ข = ພ พ v.w v = V ข ||W||2 where w is a vector in the specified direction. (B) Find a and ዐ with W = ༅།། b. (vii) Evaluate the scalar triple product, c. (a × b) then multiply the result by 1/2. Give a geomet- ric/physical interpretation of the resultant value.
Transcribed Image Text:Consider the position vectors @ = 1î −3ĵ+2k and b = −1î÷2ĵ– 2 k. (i) Find the lengths of a and b. (ii) Find the unit vectors êa and ê associated respectively with a and b. (iii) Use the cross product to find a vector c that is perpendicular to the plane passing through the origin and containing a and b. Then find the associated unit vector êc. บ (iv) A set of vectors {v₁, V2, ..., V₁} is said to form an orthogonal set if the vectors are mutually orthogonal. That is, they must satisfy = 0, ij V₁ • V j # 0, i = j. If a set of unit vectors satisfies the conditions above, the vectors are said to form an orthonormal basis. A set of vectors is said to be linearly independent if the vector equation n Σ ci v₁ = 0 has only the trivial solution c₂ = 0, Vi. i=1 Show that the set of vectors {a, b, c} are linearly independent but not orthogonal. Conse- quently, what can be said about the basis {êa, êb, êc}? (v) Find the equation of the plane passing through the terminal points of the position vectors a, b, and c. (vi) (A) A vector v can be decomposed into components that are perpendicular v to some specified direction, such that v = v ₁ + || With the aid of a diagram explain the definitions and parallel v ༅༎ ข.ข ข = ພ พ v.w v = V ข ||W||2 where w is a vector in the specified direction. (B) Find a and ዐ with W = ༅།། b. (vii) Evaluate the scalar triple product, c. (a × b) then multiply the result by 1/2. Give a geomet- ric/physical interpretation of the resultant value.
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