Find a vector x⃗ which is perpendicular to both A⃗ and B⃗ but has magnitude equal to that of B⃗.

Name: Problem 83, Chapter 1: A Complete Resource Book in Physics for JEE Main
Uploaded: 2021-10-09T13:10:36-08:00
Duration: 2 min 25 s
Channel: Akshaya Rs
Description: Find a vector x⃗ which is perpendicular to both A⃗ and B⃗ but has magnitude equal to that of B⃗. Rule: Inter change coeff. of î and ĵ and change sign of one of the vectors. A⃗=3 î-2 ĵ+k̂, B⃗=4 î+3 ĵ-2 k̂ (A) (1)/(√(10))(î+10 ĵ+17 k̂) (B) (1)/(√(10))(î-10 ĵ+17 k̂) (C) √((29)/(390))(î-10 ĵ+17 k̂) (D) √((29)/(390))(î+10 ĵ+17 k̂)

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

Question

Find a vector $\vec{x}$ which is perpendicular to both $\vec{A}$ and $\vec{B}$ but has magnitude equal to that of $\vec{B}$. Rule: Inter change coeff. of $\hat{i}$ and $\hat{j}$ and change sign of one of the vectors. $\vec{A}=3 \hat{i}-2 \hat{j}+\hat{k}, \vec{B}=4 \hat{i}+3 \hat{j}-2 \hat{k}$ (A) $\frac{1}{\sqrt{10}}(\hat{i}+10 \hat{j}+17 \hat{k})$ (B) $\frac{1}{\sqrt{10}}(\hat{i}-10 \hat{j}+17 \hat{k})$ (C) $\sqrt{\frac{29}{390}}(\hat{i}-10 \hat{j}+17 \hat{k})$ (D) $\sqrt{\frac{29}{390}}(\hat{i}+10 \hat{j}+17 \hat{k})$

   Find a vector $\vec{x}$ which is perpendicular to both $\vec{A}$ and $\vec{B}$ but has magnitude equal to that of $\vec{B}$. Rule: Inter change coeff. of $\hat{i}$ and $\hat{j}$ and change sign of one of the vectors.
$\vec{A}=3 \hat{i}-2 \hat{j}+\hat{k}, \vec{B}=4 \hat{i}+3 \hat{j}-2 \hat{k}$
(A) $\frac{1}{\sqrt{10}}(\hat{i}+10 \hat{j}+17 \hat{k})$
(B) $\frac{1}{\sqrt{10}}(\hat{i}-10 \hat{j}+17 \hat{k})$
(C) $\sqrt{\frac{29}{390}}(\hat{i}-10 \hat{j}+17 \hat{k})$
(D) $\sqrt{\frac{29}{390}}(\hat{i}+10 \hat{j}+17 \hat{k})$

A Complete Resource Book in Physics for JEE Main

Sanjeev Kumar 1st Edition

Chapter 1, Problem 83

Instant Answer

Solved by Expert Akshaya Rs

10/09/2021

Step 1

The cross product is given by the determinant of the matrix formed by the unit vectors $\hat{i}$, $\hat{j}$, $\hat{k}$ and the components of $\vec{A}$ and $\vec{B}$. \[ \vec{C} = \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & -2 & 1 \\ Show more…

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Find a vector $\vec{x}$ which is perpendicular to both $\vec{A}$ and $\vec{B}$ but has magnitude equal to that of $\vec{B}$. Rule: Inter change coeff. of $\hat{i}$ and $\hat{j}$ and change sign of one of the vectors. $\vec{A}=3 \hat{i}-2 \hat{j}+\hat{k}, \vec{B}=4 \hat{i}+3 \hat{j}-2 \hat{k}$ (A) $\frac{1}{\sqrt{10}}(\hat{i}+10 \hat{j}+17 \hat{k})$ (B) $\frac{1}{\sqrt{10}}(\hat{i}-10 \hat{j}+17 \hat{k})$ (C) $\sqrt{\frac{29}{390}}(\hat{i}-10 \hat{j}+17 \hat{k})$ (D) $\sqrt{\frac{29}{390}}(\hat{i}+10 \hat{j}+17 \hat{k})$

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Key Concepts

Cross Product

The cross product is an operation on two vectors in three-dimensional space that produces a third vector which is perpendicular to both of the original vectors. This operation is fundamental in determining orthogonality and is widely used in physics and engineering to find normals to surfaces and planes.

Vector Magnitude

Vector magnitude, or norm, is a measure of the length of a vector in space. Calculating and comparing magnitudes is essential when scaling vectors or ensuring that a vector meets certain size criteria, such as having the same length as another vector.

Normalization and Scaling

Normalization is the process of adjusting a vector to have a unit length, while scaling involves multiplying a normalized vector by a desired magnitude. This combination of techniques allows us to create vectors that not only point in a specified direction but also have a predetermined length, an important concept in many areas of applied mathematics and physics.

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