Linear Algebra Let \(\mathbf{v} = \langle 2, 4, 3 \rangle\) and \(\mathbf{w} = \langle 5, 4, -4 \rangle\). Express the vector \(\mathbf{v}\) as the sum of a vector parallel to \(\mathbf{w}\) and a vector orthogonal to \(\mathbf{w}\).We denote the vector component of \(\mathbf{v}\) along \(\mathbf{w}\) as \(\mathbf{p}\), so \(\mathbf{p} = \text{proj}_{\mathbf{w}}\mathbf{v}\).We denote the vector component of \(\mathbf{v}\) orthogonal to \(\mathbf{w}\) as \(\mathbf{q}\), and have \(\mathbf{q} = \mathbf{v} - \mathbf{p}\), so that \(\mathbf{v} = \mathbf{p} + \mathbf{q}\).(a) Determine the vector component of \(\mathbf{v}\) along \(\mathbf{w}\):\[\mathbf{p} = \](b) Determine the vector component of \(\mathbf{v}\) orthogonal to \(\mathbf{w}\):\[\mathbf{q} = \]Express all answers in exact form, no decimals.

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Let = (2, 4, 3) and = (5, 4, 4). Express the vector as the sum of a vector parallel to w and a vector orthogonal to w. We denote the vector component of v along was p, so p = projv. We denote the vector component of orthogonal to was ad, and have a = - p , so that v=p+q. (a) Determine the vector component of u along w: р = (b) Determine the vector component of orthogonal to w: q = 1 Express all answers in exact form, no decimals. Submit Question

Let = (2, 4, 3) and = (5, 4, 4). Express the vector as the sum of a vector parallel to w and a vector orthogonal to w. We denote the vector component of v along was p, so p = projv. We denote the vector component of orthogonal to was ad, and have a = - p , so that v=p+q. (a) Determine the vector component of u along w: р = (b) Determine the vector component of orthogonal to w: q = 1 Express all answers in exact form, no decimals. Submit Question

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Linear Algebra

Let \(\mathbf{v} = \langle 2, 4, 3 \rangle\) and \(\mathbf{w} = \langle 5, 4, -4 \rangle\). Express the vector \(\mathbf{v}\) as the sum of a vector parallel to \(\mathbf{w}\) and a vector orthogonal to \(\mathbf{w}\).

We denote the vector component of \(\mathbf{v}\) along \(\mathbf{w}\) as \(\mathbf{p}\), so \(\mathbf{p} = \text{proj}_{\mathbf{w}}\mathbf{v}\).

We denote the vector component of \(\mathbf{v}\) orthogonal to \(\mathbf{w}\) as \(\mathbf{q}\), and have \(\mathbf{q} = \mathbf{v} - \mathbf{p}\), so that \(\mathbf{v} = \mathbf{p} + \mathbf{q}\).

(a) Determine the vector component of \(\mathbf{v}\) along \(\mathbf{w}\):

\[
\mathbf{p} =
\]

(b) Determine the vector component of \(\mathbf{v}\) orthogonal to \(\mathbf{w}\):

\[
\mathbf{q} =
\]

Express all answers in exact form, no decimals.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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