Vectors

Step 1: A vector perpendicular to the plane of $%5Cvec%7Br_1%7D%2C%20%5Cvec%7Br_2%7D$ .<
A suitable vector

%5Cvec%7Br%7D

is parallel to

%5Cvec%7Br_1%7D%5Ctimes%20%5Cvec%7Br_2%7D

. First find

%5Cvec%7Br_1%7D%5Ctimes%20%5Cvec%7Br_2%7D%20%3D%20%5Cbegin%7Bvmatrix%7D%20%5Cmathbf%7Bi%7D%20%26%20%5Cmathbf%7Bj%7D%20%26%20%5Cmathbf%7Bk%7D%3Cbr%3E%202%20-1%200%3Cbr%3E%200%201%202%20%5Cend%7Bvmatrix%7D.

Compute:

%3D%20%5Cmathbf%7Bi%7D%5Cbigl((-1)%5Ccdot2%20-0%5Ccdot1%5Cbigr)%20-%5Cmathbf%7Bj%7D%5Cbigl(2%5Ccdot2%20-0%5Ccdot0%5Cbigr)%20%2B%5Cmathbf%7Bk%7D%5Cbigl(2%5Ccdot1%20-(-1)%5Ccdot0%5Cbigr).

%3D%20%5Cmathbf%7Bi%7D%5C%2C(-2)%20-%5Cmathbf%7Bj%7D%5C%2C(4)%20%2B%5Cmathbf%7Bk%7D%5C%2C(2).

%5Cvec%7Br_1%7D%5Ctimes%20%5Cvec%7Br_2%7D%20%3D%20-2%5C%2C%5Cmathbf%7Bi%7D%20-4%5C%2C%5Cmathbf%7Bj%7D%20%2B%202%5C%2C%5Cmathbf%7Bk%7D.

Step 2: Let $%5Cvec%7Br%7D$ be a scalar multiple.
Thus

%5Cvec%7Br%7D%3D%5Clambda%5C%2C(-2%5C%2C%5Cmathbf%7Bi%7D%20-4%5C%2C%5Cmathbf%7Bj%7D%20%2B2%5C%2C%5Cmathbf%7Bk%7D).

Step 3: Projection on $%5Cvec%7Bv%7D%3D2%5Cmathbf%7Bi%7D%2B%5Cmathbf%7Bj%7D%2B2%5Cmathbf%7Bk%7D$ .
The magnitude of the projection of

%5Cvec%7Br%7D

onto

%5Cvec%7Bv%7D

%5Ctext%7Bproj%7D_%7B%5Cvec%7Bv%7D%7D(%5Cvec%7Br%7D)%20%3D%20%5Cfrac%7B%5Clvert%20%5Cvec%7Br%7D%5Ccdot%20%5Cvec%7Bv%7D%5Crvert%7D%7B%5Clvert%5Cvec%7Bv%7D%5Crvert%7D.

Weâ€™re given this is

1

. Now

%5Cvec%7Br%7D%5Ccdot%20%5Cvec%7Bv%7D%20%3D%20%5Clambda%5C%2C(-2%5C%2C%5Cmathbf%7Bi%7D%20-4%5C%2C%5Cmathbf%7Bj%7D%20%2B2%5C%2C%5Cmathbf%7Bk%7D)%20%5C%2C%5Ccdot%5C%2C(2%5C%2C%5Cmathbf%7Bi%7D%2B%5Cmathbf%7Bj%7D%2B2%5C%2C%5Cmathbf%7Bk%7D)

%3D%20%5Clambda%5CBigl%5B%20(-2)%5Ccdot2%20%2B(-4)%5Ccdot1%20%2B%20(2)%5Ccdot2%20%5CBigr%5D%20%3D%20%5Clambda%5CBigl%5B-4%20-4%20%2B4%5CBigr%5D%20%3D%20%5Clambda%5Ccdot(-4).

%7C%5Cvec%7Br%7D%5Ccdot%20%5Cvec%7Bv%7D%7C%3D%20%7C%5C%2C%5Clambda%5Ccdot(-4)%5C%2C%7C%3D4%7C%5Clambda%7C.

Next,

%5Clvert%20%5Cvec%7Bv%7D%5Crvert%3D%5Csqrt%7B2%5E2%20%2B1%5E2%20%2B2%5E2%7D%3D%5Csqrt%7B4%2B1%2B4%7D%3D%20%5Csqrt%7B9%7D%3D3.

Hence:

%5Ctext%7Bproj%7D_%7B%5Cvec%7Bv%7D%7D(%5Cvec%7Br%7D)%20%3D%20%5Cfrac%7B4%7C%5Clambda%7C%7D%7B3%7D%20%3D1%20%5C%3B%5C%3B%5CLongrightarrow%5C%3B%5C%3B%20%7C%5Clambda%7C%20%3D%5Cfrac%7B3%7D%7B4%7D.

We can assume

%5Clambda%20%3E0

if direction is not specifically reversed. So

%5Clambda%3D%5Ctfrac%7B3%7D%7B4%7D

. Step 4: $%5Clvert%20%5Cvec%7Br%7D%5Crvert$ .

%5Cvec%7Br%7D%3D%5Ctfrac34(-2%5C%2C%5Cmathbf%7Bi%7D-4%5C%2C%5Cmathbf%7Bj%7D%2B2%5C%2C%5Cmathbf%7Bk%7D).

Hence

%5Clvert%20%5Cvec%7Br%7D%5Crvert%20%3D%20%5Cleft%7C%5Ctfrac%7B3%7D%7B4%7D%5Cright%7C%5C%3B%5Csqrt%7B(-2)%5E2%2B(-4)%5E2%2B(2)%5E2%7D%20%3D%20%5Cfrac%7B3%7D%7B4%7D%5C%3B%5Csqrt%7B4%2B16%2B4%7D%20%3D%20%5Cfrac%7B3%7D%7B4%7D%5C%3B%5Csqrt%7B24%7D%20%3D%20%5Cfrac%7B3%7D%7B4%7D%5Ctimes%202%5Csqrt%7B6%7D%20%3D%20%5Cfrac%7B3%5Csqrt%7B6%7D%7D%7B2%7D.

Therefore

%5Cboxed%7B%5Ctfrac%7B3%5Csqrt%7B6%7D%7D%7B2%7D%7D

Step 1: Compute $%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D$ . $%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D%20%3D%20%5Cbegin%7Bvmatrix%7D%20%5Cmathbf%7Bi%7D%20%26%20%5Cmathbf%7Bj%7D%20%26%20%5Cmathbf%7Bk%7D%20%5C%5C%201%20%26%20-1%20%26%202%20%5C%5C%201%20%26%202%20%26%20-1%20%5Cend%7Bvmatrix%7D$ $%3D%20%5Cmathbf%7Bi%7D%5Cbigl((-1)%5Ccdot(-1)%20-%202%5Ccdot2%5Cbigr)%20-%20%5Cmathbf%7Bj%7D%5Cbigl(1%5Ccdot(-1)%20-2%5Ccdot1%5Cbigr)%20%2B%20%5Cmathbf%7Bk%7D%5Cbigl(1%5Ccdot2%20-(-1)%5Ccdot1%5Cbigr).$ $%3D%20%5Cmathbf%7Bi%7D%5C%2C(1-4)%20-%5Cmathbf%7Bj%7D%5C%2C(-1%20-2)%20%2B%5Cmathbf%7Bk%7D%5C%2C(2%2B1)$ $%3D%20-3%5C%2C%5Cmathbf%7Bi%7D%20%2B3%5C%2C%5Cmathbf%7Bj%7D%20%2B3%5C%2C%5Cmathbf%7Bk%7D.$ So $%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D%3D3(-%5Cmathbf%7Bi%7D%20%2B%5Cmathbf%7Bj%7D%20%2B%5Cmathbf%7Bk%7D)$ .

Step 2: Use $%5Ccos%5Cangle(%5Cvec%7Ba%7D%2C%5C%2C%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D)%3D%5Ctfrac%7B2%7D%7B%5Csqrt%7B3%7D%7D$ .

Since $%5Cvec%7Ba%7D$ is a unit vector, we compute: $%5Clvert%20%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D%20%5Crvert%20%3D%20%5Clvert%203(-%5Cmathbf%7Bi%7D%20%2B%5Cmathbf%7Bj%7D%20%2B%5Cmathbf%7Bk%7D)%20%5Crvert$ $%3D%203%5Csqrt%7B3%7D.$ Using the formula: $%5Ccos%5Cangle(%5Cvec%7Ba%7D%2C%5C%2C%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D)%20%3D%20%5Cfrac%7B%5Cvec%7Ba%7D%5Ccdot(%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D)%7D%7B%5Clvert%20%5Cvec%7Ba%7D%5Crvert%5C%2C%5Clvert%20%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D%5Crvert%7D$ We solve for: $%5Cvec%7Ba%7D%5Ccdot(%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D)%20%3D%206.$

Step 3: Find $%5Clvert%20%5Cvec%7Ba%7D%5Ctimes(%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D)%5Crvert$ .

Using the vector triple product identity: $%5Clvert%20%5Cvec%7Ba%7D%5Ctimes(%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D)%5Crvert%5E2%20%2B%20(%5Cvec%7Ba%7D%5Ccdot(%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D))%5E2%20%3D%20%5Clvert%20%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D%5Crvert%5E2.$ Substituting values: $%5Clvert%20%5Cvec%7Ba%7D%5Ctimes(%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D)%5Crvert%5E2%20%2B%206%5E2%20%3D%20(3%5Csqrt%7B3%7D)%5E2.$ $%5Clvert%20%5Cvec%7Ba%7D%5Ctimes(%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D)%5Crvert%5E2%20%3D%2027%20-%2036%20%3D%209.$

Thus, the final result is:

$%5Clvert%20%5Cvec%7Ba%7D%5Ctimes(%5Cvec%7Bb%7D%5Ctimes%20%5Cvec%7Bc%7D)%5Crvert%20%3D%20%5Cboxed%7B3%7D.$

About Vectors - TS-EAMCET

Vectors is a vital chapter for TS-EAMCET aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Vectors PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Vectors carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
6 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

About Vectors - TS-EAMCET

Frequently Asked Questions

Why focus on Vectors PYQs?

How to best use this analysis?

Vectors

High-Yield Trend

Chapter Questions 6 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

About Vectors - TS-EAMCET

Frequently Asked Questions

Why focus on Vectors PYQs?

How to best use this analysis?

Chapter Questions
6 MCQs