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Question #41930

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{v}_k = \frac{k}{2} \hat{i} + |k-5| \hat{j}$ and $\vec{u}_k = \hat{i} + (-1)^k \hat{j}$ be two vectors for $k \in \mathbb{N}$ . If $P_k$ is the scalar projection of $\vec{v}_k$ onto $\vec{u}_k$ , evaluate the value of: $\left[ \sum_{k=1}^{21} \sqrt{2} P_k \right] - \sum_{k=1}^{21} \left[ \sqrt{2} P_k \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$ .

Options

A.
5
B.
-3
C.
6
D.
0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Dot Product Projection Sign|Sigma Notation Properties|Integral Part and Fractional Part|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the non-linear properties of the greatest integer function $[x]$ and fractional part $\{x\}$ when applied to a summation of vector projections.

The core trap lies in the evaluation of $\{y_k\}$ for negative values.

Students frequently make the error of assuming $\{-x\} = -\{x\}$ , which would lead them to incorrectly assign a fractional part of $-0.5$ instead of $+0.5$ for terms where the projection is negative (e.g., $k=1 \implies y_1 = -3.5 \implies \{-3.5\} = 0.5$ ).

Furthermore, it tests the conceptual understanding that $[\sum y_k] - \sum [y_k] = \sum \{y_k\} - \{\sum y_k\}$ , preventing brute-force calculation of all 21 floor terms and rewarding algebraic manipulation of periodic functions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Dot Product Projection Sign
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free