Given A⃗=2 î+p ĵ+q k̂ and B⃗=5 î+7 ĵ+3 k̂. If A⃗ B⃗ then the values of p and q are, respectivel

step 1 In this question we will learn about some basic concept of the vectors. The question is we have

Given A⃗=2 î+p ĵ+q k̂ and B⃗=5 î+7 ĵ+3 k̂. If A⃗ B⃗ then...

Given $\vec{A}=2 \hat{i}+p \hat{j}+q \hat{k}$ and $\vec{B}=5 \hat{i}+7 \hat{j}+3 \hat{k}$. If $\vec{A} \| \vec{B}$ then the values of $p$ and $q$ are, respectively, (A) $\frac{14}{5}$ and $\frac{6}{5}$ (B) $\frac{14}{3}$ and $\frac{6}{5}$ (C) $\frac{6}{5}$ and $\frac{1}{3}$ (D) $\frac{3}{4}$ and $\frac{1}{4}$

Given $\vec{A}=2 \hat{i}+p \hat{j}+q \hat{k}$ and $\vec{B}=5 \hat{i}+7 \hat{j}+3 \hat{k}$. If $\vec{A} \| \vec{B}$
then the values of $p$ and $q$ are, respectively,
(A) $\frac{14}{5}$ and $\frac{6}{5}$
(B) $\frac{14}{3}$ and $\frac{6}{5}$
(C) $\frac{6}{5}$ and $\frac{1}{3}$
(D) $\frac{3}{4}$ and $\frac{1}{4}$

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Key Concepts

Component-wise Comparison

When two vectors are set to be parallel, each corresponding component of the first vector must equal the scalar multiple of the corresponding component of the second vector. This leads to a system of equations that can be solved to find any missing or unknown components, making this method fundamental in many vector problems.

Vector Parallelism

This concept refers to the condition where two vectors lie along the same line or direction, meaning one is a scalar multiple of the other. In vector algebra, if vectors are parallel, the ratio of their corresponding components is constant, which forms the basis for setting up equations to find unknown values in the vectors.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a constant, which scales its magnitude without changing its direction. When establishing that two vectors are parallel, one vector is expressed as a scalar multiple of the other, allowing the unknown components to be determined by comparing the constants across all dimensions.

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