Consider a vector 𝐅=4 𝐢̂-3 𝐣̂. Another vector that is perpendicular to 𝐅 is (a) 4 î+3 ĵ (b) 6 ĵ

step 1 So the question says, consider vector f equals 4i minus 3J. Another vector that is perpendicular

Consider a vector 𝐅=4 𝐢̂-3 𝐣̂. Another vector that is...

Consider a vector $\mathbf{F}=4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}$. Another vector that is perpendicular to $\mathbf{F}$ is
(a) $4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}$
(b) $6 \hat{\mathrm{j}}$
(c) $7 \hat{j}$
(d) $3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}$

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Key Concepts

Vector Components

This concept involves expressing vectors in terms of their individual components along defined basis directions, typically i? and j? in two-dimensional spaces. Understanding how to break a vector down into its components allows one to perform vector operations like dot products, which are essential for determining relationships such as perpendicularity between vectors.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar, calculated by multiplying corresponding components and summing the results. It serves as a measure of the similarity in direction between the two vectors. Crucially, if the dot product of two non-zero vectors is zero, this indicates that the vectors are perpendicular (orthogonal) to each other.

Orthogonality

Orthogonality between vectors means that they are perpendicular, which can be tested using the dot product. When two vectors are orthogonal, their dot product equals zero. This key concept in vector analysis is widely used in various fields, including physics, engineering, and mathematics, to analyze and solve problems involving perpendicular directions or independent dimensions.

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