Question

Determine whether the given vectors are orthogonal. parallel, or neither. (a) $a=(-5,3,7), \quad b=(6,-8,2)$ (b) $a=(4,6), \quad b=(-3,2)$ (c) $a=-i+2 j+5 k, \quad b=3 i+4 j-k$ (d) $a=2 i+6 j-4 k, \quad b=-3 i-9 j+6 k$

Name: Problem 21, Chapter 9: Calculus
Uploaded: 2021-09-05T17:07:42-08:00
Duration: 1 min 5 s
Channel: Carson Merrill
Description: Determine whether the given vectors are orthogonal. parallel, or neither. (a) a=(-5,3,7), b=(6,-8,2) (b) a=(4,6), b=(-3,2) (c) a=-i+2 j+5 k, b=3 i+4 j-k (d) a=2 i+6 j-4 k, b=-3 i-9 j+6 k

   Determine whether the given vectors are orthogonal. parallel, or neither.
(a) $a=(-5,3,7), \quad b=(6,-8,2)$
(b) $a=(4,6), \quad b=(-3,2)$
(c) $a=-i+2 j+5 k, \quad b=3 i+4 j-k$
(d) $a=2 i+6 j-4 k, \quad b=-3 i-9 j+6 k$

Calculus

James Stewart 4th Edition

Chapter 9, Problem 21

Instant Answer

Solved by Expert Carson Merrill

09/05/2021

Step 1

The dot product of two vectors a and b is given by the formula a.b = a1*b1 + a2*b2 + a3*b3 for 3-dimensional vectors and a.b = a1*b1 + a2*b2 for 2-dimensional vectors. Show more…

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Determine whether the given vectors are orthogonal. parallel, or neither. (a) $a=(-5,3,7), \quad b=(6,-8,2)$ (b) $a=(4,6), \quad b=(-3,2)$ (c) $a=-i+2 j+5 k, \quad b=3 i+4 j-k$ (d) $a=2 i+6 j-4 k, \quad b=-3 i-9 j+6 k$

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Key Concepts

Parallel Vectors

Two vectors are said to be parallel if one is a scalar multiple of the other. This means that each corresponding component of one vector is proportional to the corresponding component of the other, indicating that they lie along the same or exactly opposite direction.

Component Analysis

Component analysis involves examining the individual entries (coordinates) of vectors in their respective dimensions. It is fundamental for calculating operations such as the dot product and verifying if the ratios of the components are consistent, which helps in determining relationships like orthogonality or parallelism.

Dot Product

The dot product is a key operation in vector algebra where the corresponding components of two vectors are multiplied and then summed. It is used to determine the relationship between vectors; in particular, if the dot product is zero, the vectors are orthogonal.

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