Let a unit vector u = xi + yj + zk make angles (pi/2), (pi/3) and (2pi/3) with the vectors (1/sqrt(2))i + (1/sqrt(2))k, (1/sqrt(2))j + (1/sqrt(2))k and (1/sqrt(2))i + (1/sqrt(2))j respectively. If v = (1/sqrt(2))i +(1/sqrt(2))j +(1/sqrt(2))k, then |u -v |2 is equal to

Let the given baseline vectors be p1, p2, p3. Note that |p1| = |p2| = |p3| = 1. 1. Angle with p1 is (pi/2): 2. Angle with p2 is (pi/3): 3. Angle with p3 is (2pi/3): Subtracting (ii) from (iii): Solving (i) and (iv): From (ii): y = (1/sqrt(2)) - (1/sqrt(2)) = 0. So, u = -(1/sqrt(2))i + 0j + (1/sqrt(2))k. Given v = (1/sqrt(2))i + (1/sqrt(2))j + (1/sqrt(2))k: Evaluating the squared magnitude: Setting up dot products systematically transforms descriptive geometric angles into solvable sets of linear equations. Class 12 Mathematics: Vector Algebra

Let a unit vector u = xi + yj + zk make angles (pi/2), (pi/3) and (2pi/3) with the vectors (1/sqrt(2))i + (1/sqrt(2))k, (1/sqrt(2))j + (1/sqrt(2))k and (1/sqrt(2))i + (1/sqrt(2))j respectively. If v =

Let the given baseline vectors be p1, p2, p3. Note that |p1| = |p2| = |p3| = 1. 1. Angle with p1 is (pi/2): 2. Angle with p2 is (pi/3): 3. Angle with p3 is (2pi/3): Subtracting (ii) from (iii): Solving (i) and (iv): From (ii): y = (1/sqrt(2)) - (1/sqrt(2)) = 0. So, u = -(1/sqrt(2))i + 0j + (1/sqrt(2))k. Given v = (1/sqrt(2))i + (1/sqrt(2))j + (1/sqrt(2))k: Evaluating the squared magnitude: Setting up dot products systematically transforms descriptive geometric angles into solvable sets of linear

Let a unit vector u = xi + yj + zk make angles (pi/2), (pi/3) and (2pi/3) with the vectors (1/sqrt(2))i + (1/sqrt(2))k, (1/sqrt(2))j + (1/sqrt(2))k and (1/sqrt(2))i + (1/sqrt(2))j respectively. If v = (1/sqrt(2))i +(1/sqrt(2))j +(1/sqrt(2))k, then |u -v |2 is equal to

Let the given baseline vectors be p1, p2, p3. Note that |p1| = |p2| = |p3| = 1. 1. Angle with p1 is (pi/2): 2. Angle with p2 is (pi/3): 3. Angle with p3 is (2pi/3): Subtracting (ii) from (iii): Solving (i) and (iv): From (ii): y = (1/sqrt(2)) - (1/sqrt(2)) = 0. So, u = -(1/sqrt(2))i + 0j + (1/sqrt(2))k. Given v = (1/sqrt(2))i + (1/sqrt(2))j + (1/sqrt(2))k: Evaluating the squared magnitude: Setting up dot products systematically transforms descriptive geometric angles into solvable sets of linear equations. Class 12 Mathematics: Vector Algebra

Let a unit vector u = xi + yj + zk make angles (pi/2), (pi/3) and (2pi/3) with the vectors (1/sqrt(2))i + (1/sqrt(2))k, (1/sqrt(2))j + (1/sqrt(2))k and (1/sqrt(2))i + (1/sqrt(2))j respectively. If v =

Let the given baseline vectors be p1, p2, p3. Note that |p1| = |p2| = |p3| = 1. 1. Angle with p1 is (pi/2): 2. Angle with p2 is (pi/3): 3. Angle with p3 is (2pi/3): Subtracting (ii) from (iii): Solving (i) and (iv): From (ii): y = (1/sqrt(2)) - (1/sqrt(2)) = 0. So, u = -(1/sqrt(2))i + 0j + (1/sqrt(2))k. Given v = (1/sqrt(2))i + (1/sqrt(2))j + (1/sqrt(2))k: Evaluating the squared magnitude: Setting up dot products systematically transforms descriptive geometric angles into solvable sets of linear

Let a unit vector $hatmathbfu = xhatmathbfi + yhatmathbfj + zhatmathbfk$ make angles $fracpi2, fracpi3$ $\frac{\pi}{2}, \frac{\pi}{3}$ and $frac2pi3$ $\frac{2pi}{3}$ with the vectors $frac1sqrt2hatmathbfi + frac1sqrt2hatmathbfk$ , $frac1sqrt2hatmathrmj + frac1sqrt2hatmathrmk$ and $frac1sqrt2hatmathbfi + frac1sqrt2hatmathrmj$ respectively. If \vec{\mathrm{v}} = \frac{1}{\sqrt{2}}\hat{\mathrm{i}} +\frac{1}{\sqrt{2}}\hat{\mathrm{j}} +\frac{1}{\sqrt{2}}\hat{\mathrm{k}}, then |\hat{\mathbf{u}} -\bar{\mathbf{v}} |^2 is equal to

A. \frac{11}{2}
B. \frac{5}{2}
C. 9
D. 7

Solution & Explanation

### Related Formula

vecA cdot vecB = |vecA| |vecB| cos phi

### Core Logic Let the given baseline vectors be

vecp_1, vecp_2, vecp_3

. Note that

|vecp_1| = |vecp_2| = |vecp_3| = 1

. 1. Angle with

vecp_1

fracpi2

hatu cdot vecp_1 = 0 implies fracxsqrt2 + fraczsqrt2 = 0 implies x + z = 0 quad dots (i)

2. Angle with

vecp_2

fracpi3

hatu cdot vecp_2 = cosfracpi3 implies fracysqrt2 + fraczsqrt2 = frac12 implies y + z = frac1sqrt2 quad dots (ii)

3. Angle with

vecp_3

frac2pi3

hatu cdot vecp_3 = cosfrac2pi3 implies fracxsqrt2 + fracysqrt2 = -frac12 implies x + y = -frac1sqrt2 quad dots (iii)

### Step 1: Finding Vector Coordinates Subtracting (ii) from (iii):

(x + y) - (y + z) = -frac1sqrt2 - frac1sqrt2 implies x - z = -sqrt2 quad dots (iv)

Solving (i) and (iv):

2x = -sqrt2 implies x = -frac1sqrt2

z = frac1sqrt2

From (ii):

y = frac1sqrt2 - frac1sqrt2 = 0

. So,

hatu = -frac1sqrt2hati + 0hatj + frac1sqrt2hatk

. ### Step 2: Evaluating the Norm Difference Given

vecv = frac1sqrt2hati + frac1sqrt2hatj + frac1sqrt2hatk

hatu - vecv = left(-frac1sqrt2 - frac1sqrt2right)hati + left(0 - frac1sqrt2right)hatj + left(frac1sqrt2 - frac1sqrt2right)hatk

hatu - vecv = -sqrt2hati - frac1sqrt2hatj + 0hatk

Evaluating the squared magnitude:

|hatu - vecv|^2 = (-sqrt2)^2 + left(-frac1sqrt2right)^2 = 2 + frac12 = frac52

### Pattern Recognition Setting up dot products systematically transforms descriptive geometric angles into solvable sets of linear equations. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Vector Algebra

Reference Study Guides

Vector Algebra

We Map Every Repeating Question in Competitive Exams.

Say goodbye to generic mock test fatigue. RankBit uses smart analysis to group past exam questions into their foundational Repeating Question Types. Find chapter weightage, track repeating questions, and score higher with targeted practice.

Select Your Target Exam

Choose an exam track below to find formulas per chapter and patterns.

Syncing Exam Intelligence

Mapping formulas and patterns across all tracks…

PATH A — FULL LENGTH PRACTICE

Full Mock Test Hub

Simulate real NTA exam conditions with fully tracked mocks. Time yourself against past papers.

Under Development

PATH B — TARGETED PRACTICE

Topic-wise Practice Hub

Practice past-year questions one chapter at a time. Pick an exam → subject → chapter and get every PYQ for that topic — pulled together from all past papers — with the chapter's key formulas alongside.

Loading Questions... Browse Topics

Latest from the Blog

View all →

Loading articles...

Review & Tools

Mistakes To Revise

No conceptual mistakes detected.

🔥 Practice Session Live

Showing All Questions

Exam Questions

Test your concepts live. Choose options or enter numerical values, then verify your answer to reveal the double-box solution matrix.

Solution & Explanation

Reference Study Guides

More Vector Algebra Previous-Year Questions — Page 7

Solution

More Vector Algebra Questions — jee_main_2024_29_january_evening

We Map Every Repeating Question in Competitive Exams.

Select Your Target Exam

Full Mock Test Hub

Topic-wise Practice Hub

JEE Main Exam Papers

Chapter Weightage Board

Formula Recognition

Exam Questions

Select an Exam